![takes values in a separable complete metric space T; the subscript e is
a parameter which for definiteness is taken to be in (0, SQ] .
Denote by val(e) the optimal value of (1.1), namely the supremum
of the payoff integral in (1.1) subject to the given constraints.
The dependence of the problem on the parameter e is the subject of
the present paper. In particular, we wish to examine the convergence
of val(e), and the convergence of optimal solutions of (1.1) as e —• 0.
The parameter e affects the constraint set-valued map F, which takes
closed subsets of Rn
as values. (At the expense of complicated no
tations we could consider a value function u which depends on the
parameter, see Remark 6.1; see Remark 6.2 for other parameter depen
dence forms.) The technical assumptions on the data are displayed in
the next section.
Without the presence of the parameter function pe(-) the optimiza
tion problem is a classical allocation problem encountered, for instance,
in operations research and economics. The underlying space, which for
convenience is chosen here to be the interval [0,1], is interpreted as a
space of, say, agents, and the real valued function u indicates the utility
of allocating the amount x(t) to the agent t. There is the quantity à
to allocate, and the assignment of the (infinitesimal) quantity x(t) to
the agent i, results in the (infinitesimal) utility t¿(í, x(t)). Hence the
goal is to maximize the total utility, subject to the constraint à on the
resources, and subject to the constraints on the allocation represented
by F. For an analysis and applications of the allocation problem with
out the presence of the parameter e see e.g., Yaari [27], Aumann and
Perles [8], Berliocchi and Lasry [15], Artstein [2], Aumann and Shapley
[9], Arrow and Radner [1].
The e appearing in the model (1.1) signifies a possible sensitive
dependence of the problem's data on some, say small, parameter. The
sensitivity of the dependence is depicted by, possibly, highly oscillatory
behavior of p£(>) for e small. We are interested in the determination
of the limit behavior of the values of (1.1) and its optimal solutions as
e tends to 0. To this end we produce a variational limit, namely an
2](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-15-320.jpg)
![allocation problem associated with the limit as e tends to 0. The term
"variational limit" refers to the situation where the optimal value of
the limit problem is indeed the limit of the optimal value of (1.1) as e
tends to 0, and the solution of the limit problem is, in the appropriate
sense, the limit of solutions of (1.1) as e tends to 0. Furthermore, we
show how the solution of the limit problem can be employed to produce
near optimal solutions of the problem (1.1) when e is small enough.
Variational limits of optimization and control problems, with roles
similar to the description in the preceding paragraph, were identified
in a variety of problems. The T-convergence plays a similar role, see
e.g., Dal Maso [18] and references therein. It was applied to control
problems which include the form (1.1), see e.g., Buttazzo and Cavazzuti
[17], Belloni, Buttazzo and Preddi [14], and Freddi [19, 20]. The related
notion of epi-convergence is also designed to capture the limit behavior,
see Attouch [6], Rockafellar and Wets [23]. For control problems, [4]
produces variational limits based on the Young measure limits of the
parameter function Pe(-). (Young measure solutions, also called relaxed
solutions, to variational and control problems have been known for a
long time; see e.g., Young [28], Warga [26], Valadier [25], Balder [11],
Ball and Knowles [13]. It is the Young measure limit of the parameters
that is the novelty in [4].) In this paper we employ the Young mea
sure limit of the parameters for the parameterized problem (1.1). This
results in a variational limit with a multimeasure constraint set. The
purpose of this paper is to demonstrate how such a limit is generated
and how to exploit its variational limit characteristics.
The rest of the paper is organized as follows. In Section 2 we provide
the technical assumptions on the data and recall some preliminary facts,
primarily on Young measures. Section 3 applies the limit process to the
parameter functions pe and (with the aid of a result in [5]) singles out
the variational problem which arises when the limit is taken. This limit
problem has measure-valued constraints and measure valued solutions.
In Section 4 we establish, under appropriate conditions, two ingredients
of the variational limit, namely the convergence of the values and the
convergence of the solutions. We find that the convergence of solutions
3](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-16-320.jpg)
![holds is in the weak or the narrow topology. The third ingredient,
namely the generation of near optimal solutions, is examined in Section
5. Some comments on the model, in particular on the need for some
of the conditions, are listed in Section 6. In the closing section we
indicate, via an illustrative example, how the limit problem can be
analyzed and applied.
2. Preliminaries
In this section we display the technical assumptions on the data ap
pearing in the problem (1.1), including assumptions on the growth of
the utility function u and the constraints map F. We also recall some
facts concerning the narrow convergence of Young measures.
The choice of the interval [0,1] as the space underlying the inte
gration is made for convenience of notations; any complete separable
metric space with an atomless underlying probability measure on it
will do (see also Remark 6.3 for the possibility of using a measure
space without a topology).
Whenever encountered with a measurability structure on a metric
space, the Borel field is considered. We restrict the discussion to de
cisions in the non negative orthant i?+ of i?n
, namely x is assumed
greater than or equal to 0 coordinate-wise (see Remark 6.4). Likewise,
the inequality in (1.1) is interpreted coordinate-wise. We write x > 0
if each coordinate of x is strictly bigger than 0. We write x for the
euclidean norm of x. The Lebesgue measure on [0,1] is denoted by À.
For each fixed e the mapping pe(-) maps [0,1] into a given separable
and complete metric space Y (and we denote the elements in T by 7).
It is assumed that for each e the function p£(-) is measurable.
Assumption 2.1. (i) The function u (which is defined on [0,1] x
R)j is assumed measurable with respect to the variable t and con
tinuous with respect to x. (ii) The function u is o(x) as x -+ 00
integrably in t (namely, for each 6 > 0, there exists an integrable func
tion rj(t) such that u(tjx) < 6x whenever x > rj(t)). (Hi) The
4](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-17-320.jpg)
![valúes ¡Q u(t,x(t))dt are uniformly bounded for families #(•) which are
bounded by an integrable function.
The measurability and continuity assumptions on u are standard as
sumptions, yet for many applications it suffices to assume upper semi-
continuity in x. This is not the case in the present paper, as demon
strated in Example 6.5. The growth condition o{x) was introduced by
Aumann and Pedes [8, Definition 1] to guarantee existence of solutions
to the problem (1.1) with the constraint set F being equal to Rj¡_. It
is possible to show (see Remark 6.6) that under Assumption 2.1 for
every e > 0 the problem (1.1) has a solution, say #£(-), provided that
the problem is feasible, namely that there exists a measurable selection
#(•) of F(-,p£(-)) such that ¡Q x(t)dt < â. We are interested in the
limit of this solution as e tends to 0. The growth condition plays a role
in our analysis beyond guaranteeing the existence (see Example 6.7).
In what follows we consider the space of closed subsets of Z£n
, de
noted by C(Rn
); we are actually employing C(Rr
¡_). We consider C(Rn
)
with the Kuratowski convergence, namely C = Lim C¿ if every cluster
point of a sequence qfc G C¿fc is in C, and each point c in C is the limit
of a sequence c¿ in C¿. The Kuratowski convergence on closed subsets
of Rn
is metrizable, and the metric is separable and complete (see, e.g.,
[21]).
The constraints set-valued mapping F(i, 7) is defined on [0,1] x T
and takes closed subsets of i?+ as values; namely, its values are in
C(Rn
). We assume that F is measurable in the variable t and con
tinuous in the variable 7. For a fixed e the integral of the set-valued
map F(-,pe(-)) forms the feasible set for the optimization (1.1). We
postpone the discussion of the feasible set to Section 4, where we define
also the analogous set for the limit problem.
The following assumption is made in order to avoid some pathologies
(see Example 6.8):
Assumption 2.2. The family of functions
¿£(t) = min{x : x G F(t,pe(t))}
5](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-18-320.jpg)
![is uniformly integrable.
Next we recall the notions of Young measure and narrow conver
gence. For a general overview and applications see Balder [11], Val-
adier [25]. Let S be a complete separable metric space. We denote by
V(S) the space of probability measures on 5, endowed with the metric
of weak convergence of measures; the space V(S) is then a complete
separable metric space, see Billingsley [16]. A Young measure is a mea
surable mapping, say v = */(•), from [0,1] into V(S). We denote the
space of Young measures by y = 3^([0,1], S). The mapping v can be
considered as a probability measure on [0,1] x S determined by
(2.1)
As such, elements v are subject to the metric of weak convergence on
V([0,1] x S). We consider the space ^([0,1] x S) with this metric,
which we also call the metric of narrow convergence. A useful criterion
for the narrow convergence of i/¿(-) to ^o(') is a s
follows:
(2.2)
for any bounded function r : [0,1] x S —• R which is measurable in
t and continuous in s. (The customary criterion requires (2.2) for r
bounded and continuous; it is easy then to see that (2.2) holds also
for r bounded, continuous in s and only measurable in i, this since
v(E x S) = X(E) for each E C [0,1], see e.g. [5, Lemma 2.4].)
A measurable mapping h(-) from [0,1] into S can be viewed as a
Young measure which assigns to each t the Dirac measure ^h(t) ? n a m e
l y
the probability measure supported at {h(t)}. The narrow convergence
applied to point-valued maps may result in a Young measure limit.
3. The variational limit
With the aid of the narrow limit of the parameter pe(-) in (1.1), we
exhibit in this section a candidate for a variational limit for (1.1). We
6](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-19-320.jpg)
![indicate how the suggested limit arises naturally, however, after over
coming a conceptual problem, which is resolved with the aid of a result
established in [5]. In the next two sections we show that indeed a
variational limit was obtained.
Assumption 3.1. The functions p£(-) appearing in (1.1) converge
narrowly to a Young measure, say to /i(-) = /x.
See Remark 6.9 on the extent of the assumption. Next we display
the variational limit, then define and explain the terms it refers to.
(3.1)
where M(t) is the collection of probability measures on RJ. selec-
tionable with respect to the probability measure on S(R) given by
F(t,/x(t)).
In analogy with the problem (1.1) we denote by val(O) the optimal
value of (3.1).
A solution of (3.1) should be a Young measure £ = £(•) with values
in V{Rn
). Thus, the form of (3.1) is as indicated in the title of the
paper. We explain now the terms used in (3.1).
For the probability measure fi on T, the probability measure F(t, ¡i)
on C(R) is (for t fixed) the measure induced from /i by F(t, 7), namely
F(t,n)(A) = n({1:F(t,7)€A}).
Since F is assumed continuous in 7 it is clear that for t fixed F{t,n) is
indeed in V(C(Rn
)). It is also clear that since //(•) is a Young measure
into the space T, then the mapping F(-, //(•)) is a Young measure into
7](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-20-320.jpg)
![C(Rn
). It is true (see Lemma 4.2) that the the narrow convergence of
pe(-) to //(•) implies the narrow convergence of JP(-,pe(-)) to F(-, //(•))•
Let 1/ be a probability measure on C(Rn
), and let p be a probability
measure on Rn
. The measure p is selectionable with respect to v if there
are a probability measure space (iî, £, À), a measurable set-valued map
H from £î into C(Rn
) and a measurable point-valued map h from fi into
i?n
, such that /¿(u;) 6 #(u;) for all a;, Dh = p and D if = i/, where D/¿
and DH are the probability distributions induced by h and £T, namely
Dh(A) = ({u;:h(v)eA}),
and likewise for H.
Notation 3.2. For a probability measure v on C(i?n
), denote by
S{y) the family of probability measures p on i?n
which are selectionable
with respect to v. (Thus, in (3.1) M(t) = S(F(t, n(t))).)
Remark 3.3. The family S(u) for v in VC{RJ+), was characterized
in [3], with a slight extension offered in [5] (See also Ross [24]). The
family coincides with the collection of probability measures p satisfying
p(K)<v{{C:CDK¿Q})
for every compact set K C Rn
. In particular, S(v) is a convex set of
probability measures, and it is not difficult to show that it is closed
with respect to the weak convergence.
The heuristics behind the construction of the limit problem (3.1)
is that as e tends to 0, the optimal solutions of (1.1), say xe(-), may
converge as well; but since the parameter functions p£(-) converge only
narrowly, the most we can hope for is that xe(-) will converge narrowly.
Hence the limit should be a Young measure on i?!f.. This clearly ex
plains the expressions for the utility and for the integral constraint in
the limit problem (3.1). A natural restriction on the domain of £(t)
should actually have been
£(t)eF(t,Ai(t)) (3.2)
8](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-21-320.jpg)
![with F(t, fi(t)) being the induced probability measure as explained ear
lier. But, there is a syntactic problem with the form (3.2). Indeed, the
relation "#(•) is a selection of F(-,pe('))" required in (1.1), loses it
meaning when it comes to "£(•) is a selection of F(-, //(•))" written in
(3.2). The difficulty is in the requirement that a probability distribu
tion on points be a selection of a probability distribution on sets. The
problem was addressed in [3],[5], where the selectionable probabilities
as defined earlier were analyzed and employed.
4. The convergence
This section verifies that the optimization problem (3.1) is a variational
limit of (1.1) as e tends to 0, in the sense expressed in the following
theorem. In the next section we show how a solution of (3.1) gives
rise to near optimal solutions of (1.1) for e small. Recall that we work
under Assumptions 2.1, 2.2 and 3.1.
Theorem 4.1. Suppose that a fixed open neighborhood of the re
source vector à is in the feasible set of (1.1) for e small enough. Then
à is in the feasible set of (3.1), and the optimal values val(e) of (1.1)
converge as e tends to 0, to the optimal value val(0) of (3.1). Further
more, let #*(•) for e fixed be an optimal solution of (1.1). Then for
any sequence S{ converging to 0, there exists a subsequence Sj such that
#£.(•) converge narrowly to a solution £(•) of (3.1).
A term which is self explanatory, yet may need a formal definition, is
the feasible set. To this end recall the integral of a set-valued function,
say H from [0,1] into C(i?+), given by (see Aumann [7] and Klein and
Thompson [21]):
(4.1)
We call Jo F(t> Pe{t))dt the exact feasible set for the problem (1.1) with
e fixed. The apparent reason is that the integral contains the resources
9](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-22-320.jpg)
![vectors that can be allocated. The feasible set is the set which contains
all the vectors ÜQ such that there exists an a1
in the exact feasible set
such that af
< ÜQ. The assumptions in the theorem imply that the
feasible set contains actually an a!
independent of e such that a1
< à
(see Example 6.10 for the need of this condition). The exact feasible set
is convex ([7], [21]), but may not be closed. The feasible set is always
closed, in view of the fact that the integral of a closed-valued set-valued
map contains the extreme points of its closure (see, e.g., [2]).
In the spirit of (4.1) we define the integral of a set-valued map G{t)
into the space of probability measures on RJ+ as follows:
(4.2)
There is a slight abuse of formality in (4.2) as integrating sets of
probability measures results with a subset of Rn
. Since we do not need
in this paper to integrate the measures in their own space, the use
of (4.2) should not lead to a confusion. It is clear that (4.2) reduces
to (4.1) when G(t) contains only Dirac measures, namely probability
measures supported on a singleton.
We call ¡Q M(t)dt the exact feasible set for (3.1). Due to the con
vexity and closedness of M(t) it is easy to see that the exact feasible
set is a convex closed subset of i?+. The feasible set of (3.1) consists,
again, of the vectors CLQ such that there exists an af
in the exact feasible
set of (3.1) such that a' < aç).
The verification of the theorem will follow several observations, some
technical and some of interest for their own sake.
Lemma 4.2. Since p6(-) converge narrowly (as Young measures
into T) to //(•) it follows that F(-,pe(-)) converge (as Young measures
intoC(Rn
)) to F(-, //(•)).
Proof. The result follows since by the continuity of F in the 7
variable, any bounded continuous function r : [0,1] x C(Rn
) —
> i?, can
10](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-23-320.jpg)
![be interpreted as a continuous bounded function r : [0,1] x T —
» i?,
hence the criterion (2.2) for convergence can be verified.
The following is a key step in our proof. It follows from a more gen
eral result established in [5]. Recall that M(t) is the set of probability
measures selectionable with respect to ¡i(t).
Proposition 4.3. The ensemble S(F(-, pe(-))) of measurable selec
tions converge in the Kuratowski sense with respect to narrow conver
gence, to the ensemble <S(M(-)) of Young measures selectionable with
respect to M(-); namely, whenever x€i(-) for S{ -> 0 are measurable se
lections ofS(F(-,p£i(-))) which converge narrowly to a Young measure
£(•), then £(•) is in S(M(-)), and for every £(•) in <S(M(-)) there are
selections xe(-) of S(F(-, p€(-))) which converge narrowly as e tends to
0,to£(-).
Proof. Given Lemma 4.2, the result is a particular case of [5,
Theorem 6.1].
The narrow convergence by itself does not imply the quantitative
results sought after in the theorem. To establish these we need some es
timates which result from the assumptions imposed on the data. These
estimates are verified in the following results.
Notation. For each N > 0 and x G i?+ let N(x) be the vector
whose i-th coordinate is equal to the i-th. coordinate X{ of x if X{ < N,
and equal to N if X{ > N. In particular N(u(t, x)) = min(i¿(í, x),N).
Lemma 4.4. A family xe(-) of selections in S(F(-,p£(-))) such
that ¡Q,xe(t)dt < a for every e, forms a precompact family with respect
to the narrow convergence, namely every sequence has a subsequence
which converges narrowly.
Proof. It is enough to verify tightness of the corresponding mea
sures (see [16]). Tightness follows since the values of the functions are
in R7
^ and for any N > 0 the estimate
(4.3)
clearly holds.
n](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-24-320.jpg)
 and 6{t) = max{«(i,a;) : N < x < rj(t)} if
N < r)(t). For any given 6 > 0 such an N exists by Assumption 2.1.
12](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-25-320.jpg)
![By narrow convergence the convergence in (4.5) holds if u is replaced
by N(u), (see (2.2), and recall that u is assumed continuous in x).
But in view of Assumption 2.1 and the construction of N (i.e.. (4.8))
changing u in (4.5) to N(u) results in a deviation (in both sides) of at
most 6(1 + â). Since 6 is arbitrarily small the convergence in (4.5) is
verified. This completes the proof.
Lemma 4.6. Let h(-) and g(-) be integrable functions from [0,1]
into Rn
and let a be between 0 and 1. Then there exists a measurable
subset, say A, o/[0,1], whose Lebesgue measure is equal to a, such that
(4.9)
Proof. Without the requirement that À(A) = a, the claim fol
lows from the Liapunov Convexity Theorem. The additional condition
À(A) = a can be deduced when the same argument is applied to the
pair of functions (#(•)? 1) and (/¿(-),0).
The following result can be viewed as a generalization of Balder [10,
12].
Proposition 4.7. Let £(•) be a selection of M(-) for which (4-4)
holds. Then there exists a sequence xe(-) of selections of F(-,p£(-))
which converge narrowly to £(•) and such that
(4.10)
for every e, and furthermore.
(4.11)
Proof. By Proposition 4.3 there exist selections, say %(•) of
F(-, Pe(-)) which converge narrowly to £(•). we shall modify them so to
13](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-26-320.jpg)
![get the desired sequence xe(-). We first aim at modifying ze{-) and get
selections, say ye(-), which converge narrowly to f (•) and
(4.12)
(the last inequality is interpreted coordinate wise). To this end note
that standard selection theorems (see e.g., [22]) together with Assump
tion 2.2 imply the existence of selections of F(-, Pe(*)), say w£(-) which
are uniformly integrable. For each N > 0 define ye N(Í) = z£(t) if
|%(*)| < N &n
d Ve N(t) — w
e(t) otherwise. For every N the conver
gence in (4.7) holds when z£ replaces x£. In particular for e fixed, if N
is large enough we can deduce that
(4.13)
with 6(e) tending to 0 as £ tends to 0. If e is small enough, the narrow
distance between z£(-) and £(•) is small, hence for e small enough and N
large enough the Lebesgue measure of {t : ye jy(t) = w£(t)} is small. In
particular, a relation N = N(e) can be determined such that ye N(e){m
)
converges narrowly to £(•), and (in view of (4.13)) the claimed estimate
(4.12) holds for ¡/e(-) = y6iN(e){-).
Consider now a sequence, say g£(-) of selections of F(-,pe(-)) such
that
(4.14)
where o! < a, (the existence of such a! is guaranteed by the conditions
in Theorem 4.1). We use Lemma 4.6 to form a selection x£,a(') such
that xe,a{t) = ye{t) for t in a set of Lebesgue measure greater or equal
to (1 — a), and then when a is fixed and e is small enough the condition
(4.15)
14](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-27-320.jpg)
![original problem. In this section we indicate how to carry out such a
program with respect to (1.1) and (3.3). The suggested scheme is as
follows.
Proposition 5.1. Let £*(•) be an optimal solution of (3.1). Then
there exist selections x£(-) o/F(-,pe(-)) which satisfy ¡Q x£(t)dt < à
(namely they are feasible for (1-1)), and which converge narrowly to
£*(•). For these selections the following holds.
(5.1)
Proof. Existence of the selections x£(-) was established in Propo
sition 4.7. Prom property (4.5) of Proposition 4.5 it follows that
Jo u(t, x£(t))dt converge to val(O), and since val(O) is the limit of val(e)
as e tends to 0, the desired estimate (5.1) follows.
Remark 5.2. The existence of near optimal solutions, as stated
in the previous result, is implicit. We note that the construction of
selections x£(-) which converge narrowly to a prescribed Young measure
can be done using standard arguments in probability. The proof of
Proposition 4.7 provides an explicit way to modify such selections and
come up with the near optimal selections as in the previous proposition.
6. Comments
Remark 6.1. In the problem (1.1) the payoff function u is indepen
dent of the parameter e. It is possible to develop the theory with such
dependence; some modifications are then needed. One apparent modi
fication is to make the growth conditions on the value function, given
in Assumption 2.1, uniform in the parameter e. The second, more in
tricate, modification is to allow the limit payoff function to depend on
a measure valued parameter (in addition to having the solution itself
be a Young measure). Such considerations were developed in [4] for
control problems without parametrized constraints.
Remark 6.2. The parameter e in (1.1) enters through the param
eterized function Pe(-)> namely the constraints are given in the form
16](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-29-320.jpg)
![F(í, Pe(í)). In particular, the narrow limit is examined on the functions
Pe(-). This form was chosen for the convenience of notations only. We
note here that such a form does not reflect any loss of generality with
respect to the standard form of parameter dependence, namely using
the form Fe(t) for the parametrized constraints. The conversion from
the latter form to a representation by a mapping p£ can be done by
letting pe(t) = F£(t). Such considerations are elaborated upon in [4],
where the same equivalence is noted even for state dependent functions.
Remark 6.3. The underlying space we use for the problem (1.1)
is the unit interval, but as mentioned, any separable complete metric
space will do. It should be noted that employing the setting worked
out in Balder [11], would make it possible to consider the integration in
(1.1) on an underlying space without a topological structure, namely
on a general atomless measure space.
Remark 6.4. We follow Aumann and Perles [8] in assuming that
x is in the non-negative orthant of Rn
; it is clear, however, that this
assumption may be relaxed. In fact, all the results hold when the
values of set-valued function F are bounded from below by an integrable
function into Rn
. Some bound from below is, however, needed.
Example 6.5. For a non parametrized problem of the form (1.1)
it is enough, hence customary, to assume that u is merely upper semi-
continuous in the x variable (it is then called a normal integrand); see
Aumann and Perles [8], Berliocchi and Lasry [15]; see also Rockafellar
[22] for a general discussion on normal integrands. In the case of the
present paper the continuity of u in the x variable is needed. As a
counter example consider the case where x is real, ã = 1, u(t,x) = 1
for x = 1 and i¿(í, x) = 0 otherwise, j(t) = e and F(i, e) is [0,1 — e].
The limit problem (which does not involve measures in this case) is
defined by F(t) = [0,1], but the values do not converge. Another place
for the need of u continuous in x is when the near optimal solutions are
constructed (see Section 5). The argument there is based on the nar
row convergence of the approximations to the limit solution. Without
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![the continuity of u in the x variable there is no guarantee that a good
approximation of the payoff is obtained.
Remark 6.6. Aumann and Perles [8] established the existence
of an optimal solution of (1.1) (with e fixed) when the constraint set
F is identical to BJ+. When F is a set-valued function with values
being closed subsets of i?+, we have first to assume that there exists
at least one selection of F(-) that meets the constraints. Then the
arguments in [8] which ensure the existence apply also in the present
case. Alternatively, the existence result follows as a particular case of
[2, Theorem 1].
Example 6.7. The growth condition o(x) of Assumption 2.1(ii)
is needed not only for the existence of optimal solutions. Consider the
case where x is a scalar, à = 1, u(t,x) = x, j(t) = £, and F(t,e) =
{OjU^-^oo]. For every e an optimal solution exists and the optimal
payoff is 1. The optimal solutions even converge narrowly, yet in the
limit, the payoff is 0. The boundedness condition on JQ u(t,x(t))dt
stated in Assumption 2.1(iii) cannot be dropped, as without it val(s)
may be infinite and the continuity at 0 may not hold. This assumption
follows when the growth condition (ii) holds and u(t,x) is monotone
in x.
Example 6.8. We show that Assumption 2.2 cannot be dropped
(note that in the standard framework, e.g., as in [8], the assumption is
fulfilled in a trivial way as 0 is a feasible selection). Let x be scalar,
à = 1, t¿(í, x) = min(x, 1), 7 = £, and let F(t, e) = {0,1} if e < t < 1
and F(t,e) = [e:_1
,oo] when 0 < t < e. For e fixed, the only way
to satisfy the constraints results in a payoff equal to e. In the limit
problem a payoff 1 can be obtained.
Remark 6.9. The prime advantage of Young measures is that
compactness, namely existence of converging subsequences, holds under
mild conditions. This is the rational behind employing Young measures
as a solution concept. See Young [28], Warga [26], Valadier [25] and
Balder [11]. Compactness is equivalent to: For each 6 there exists a
compact set K in V such that ({t : p£(t) £ K}) < 6.
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![Example 6.10. We show that the existence of the vector a1
< à
in the feasible set of (1.1) for e small enough, cannot be dropped from
the assumptions of Theorem 4.1. Consider x = (#i,£2) in
R2
, let
à = (1,1), u(t, x) = min(:ri, x2), J = e, and F(t, e) = {(2,0), (0,2), (1+
£, 1 + e)}. Then à is in the feasible set of (1.1) for every e > 0, but the
payoff is 0. The optimal payoff in limit problem is clearly 1.
7. An illustrative example
This section exhibits an example that besides serving as an illustration
to the abstract result, indicates how the variational limit can be solved.
To this end we start with a characterization of solutions to the limit
variational problem. For simplicity, the conditions are stated under
assumptions on the data stronger than previously in the text.
Proposition 7.1. Suppose that the utility function u(t,x) is non-
decreasing in the variable x. Suppose also that the vector à is in the
exact feasible set, namely a selection v(-) of M(-) exists, such that
(7.1)
Then a necessary and sufficient condition for a measurable selection
*/*(•) of M(-) for which (7.1) is satisfied, to be an optimal solution of
(3.1) is the existence of a vector q > 0, such that
(7.2)
for almost every t G [0,1].
Proof. The condition (7.2) is the Kuhn-Tucker condition for the
problem (3.1). Compare with Aumann and Perles [8, Theorem 5.1].
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![here [t/q] is the largest integer which is less than or equal to t/q. The
multiplier q associated with the optimal solution is then determined by
the equality
(7.5)
The preceding equality can be made more concrete as follows. For a
given q > 0, let n = [t/q], namely n is the largest integer such that
nq < 1 and (n + l)q > 1. Then (7.5) reads as follows.
(7.6)
For a given à the solution of (7.5) (or (7.6)) is unique. Indeed, the left
hand side is increasing as q decreases.
For a given à the multiplier q can be determined through the equa
tion (7.6), and the optimal solution for the variational limit is then the
Young measure given in (7.4). Near optimal solutions for small e are
then determined by a narrow limit of this Young measure. A direct
inspection shows that a near optimal solution is given by (7.6), x£(t) is
the even or odd number among max([i/g] — 1,0) and max([t/g] — 2], 0),
according to whether sign cos(e-1
i) is +1 or — 1.
References
[1] K. J. Arrow and R. Radner, Allocation of resources in large teams.
Econometric^ 47 (1979), 361-385.
[2] Z. Artstein, On a variational problem. J. Math. Anal Appl. 45
(1974), 404-415.
[3] Z. Artstein, Distributions of random sets and random selections. Is-
rael J. Math. 46 (1983), 313-324.
[4] Z. Artstein, Chattering variational limits of control systems. Forum
Mathematicum 5 (1993), 369-403.
[5] Z. Artstein, Relaxed multifunctions and Young multimeasures. Set-
Valued Anal, to appear.
21](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-34-320.jpg)
![[6] H. Attouch, Variational Convergence of Functions and Operators.
Pitman, Boston, 1984.
[7] R. J. Aumann, Integrals of set valued functions. J. Math. Anal.
Appl. 12 (1965), 1-12.
[8] R. J. Aumann and M. A. Perles, A variational problem arising in
economics. J. Math. Anal. Appl. 11 (1965), 488-503.
[9] R. J. Aumann and L. S. Shapley, Values of Non Atomic Games,
Princeton University Press, Princeton, 1974.
[10] E. J. Balder, A general denseness result in relaxed control the
ory. Bull. Australian Math. Soc. 30 (1984), 463-465.
[11] E. J. Balder, Lectures on Young Measures. Cahiers de
Mathématiques de la Décision 9517, CEREMADE, Université
Paris-Dauphine, Paris, 1995.
[12] E. J. Balder, Consequence on denseness of Dirac Young measures. J.
Math. Anal. Appl. 207 (1997), 536-540.
[13] J. M. Ball and G. Knowles, Young measures and minimization prob
lems in mechanics. In: Elasticity, Mathematical Methods and Appli
cations, G. Eason and R. W. Ogden eds., Ellis Horwood, Chichester
1990, pp 1-20.
[14] M. Belloni, G. Buttazzo, and L. Preddi, Completion by gamma-
convergence for optimal control problems. Ann. Fac. Sci. Toulouse
Math. 2 (1993), 149-162.
[15] H. Berliocchi and J.-M. Lasry, Intégrandes normales et mesures
paramétrées en calcul des variations. Bull. Soc. Math. France 101
(1973), 129-184.
[16] P. Billingsley, Convergence of Probability Measures. Wiley, New
York, 1968.
[17] G. Buttazzo and E. Cavazzuti, Limit problems in optimal control
theory. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989),
suppl., 151-160.
[18] G. Dal Maso, Introduction to T-Convergence. Birkhauser, Boston,
1993.
22](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-35-320.jpg)
![[19] L. Freddi, Variational limits of optimal control problems. Research
Report, University of Udine, 1997.
[20] L. Preddi, T-convergence and chattering limits in optimal control
theory. Research Report, University of Udine, 1998.
[21] E. Klein and A. C. Thompson, Theory of Correspondences. Wiley-
Interscience, New York, 1984.
[22] R. T. Rockafellar, Integral functional, normal integrands and mea
surable selections. In: Nonlinear Operators and the Calculus of
Variations, L. Waelbroeck, éd., Lecture Notes in Mathematics 543,
Springer-Verlag, Berlin, 1976, 157-208.
[23] R. T. Rockafellar and R. J-B Wets, Variational Analysis, an in
troduction. In: Multifunctions and Integrands, Stochastic Analysis
and Optimization, G. Salinetti éd., Lecture Notes in Mathematics
1091, Springer-Verlag, Berlin 1984, 1-54.
[24] D. Ross, Selectionable distributions for a random set. Math. Proc.
Cambridge Phil. Soc. 108 (1990), 405-408.
[25] M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ.
Trieste 26 (1994) supp., 349-394.
[26] J. Warga, Optimal Control of Differential and Functional Equa
tions. Academic Press, New York, 1975.
[27] M. E. Yaari, On the existence of optimal plan in continuous-time
allocation process. Econometrica 32 (1964), 576-590.
[28] L. C. Young, Lectures on the Calculus of Variations and Optimal
Control Theory. Saunders, New York, 1969.
Zvi Artstein
Department of Theoretical Mathematics
The Weizmann Institute of Science
Rehovot, ISRAEL
e-address: zvika@wisdom.weizmann.ac.il
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![1
E J BALDER
New fundamentals of Young measure
convergence
Introduction
This paper presents a new, penetrating approach to Young measure convergence in
an abstract, measure theoretical setting. It was started in [12, 13, 14] and given
its definitive shape in [18, 22]. This approach is based on K-convergence, a device
by which narrow convergence on V(Rd
) can be systematically transferred to Young
measure convergence. Here ^(R^) stands for the set of all probability measures on Rd
(in the sequel, a much more general topological space S is used instead of Rd
). Recall
that in this context Young measures are measurable functions from an underlying
finite measure space (iî,,4,fi) into V(Rd
). Recall also from [12], [13] (see also [24])
that 7-convergence takes the following form when applied to Young measures (see
Definition 3.1): A sequence (6k) of Young measures K-converges to a Young measure
¿o [notation: 6k —• ¿o] if f°r
every subsequence (6^) of (6k) the following pointwise
Cesaro-type convergence takes places
at //-almost every point UJ in H. Here "=>" means classical narrow convergence on
T^R^) (see Definition 2.1). As is shown much more completely in Proposition 3.6 and
Theorem 4.8, the following fundamental relationship holds between Young measure
convergence, denoted by " = > " , and K-convergence as just defined [18, Corollary 3.16]:
Theorem 1.1 Let (6n) be a sequence of Young measures. The following are equivalent:
(a) 6n = > 60.
(b) Every subsequence (6n>) of (6n) contains a further subsequence (6n») such that
6n" — • ^o-
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![Both the nature of this equivalence result and the way in which we shall employ it
are rather reminiscent of the well-known characterization of convergence in measure in
terms of convergence almost everywhere. But while the latter result is simple, the for
mer one is deep, as will become clear in the sequel. Nevertheless, thanks to this result
several fundamental results on (sequential) Young measure convergence become simple
to derive and can be stretched to what are arguably their most general versions in an
abstract setting. These include: (1) the Prohorov-type criterion for relative sequential
narrow compactness (Theorem 4.10), (2) the support theorem (Theorem 4.12, (3) the
lower closure theorem (Theorem 4.13), (4) the denseness theorem for Dirac Young
measures. The power of the apparatus thus developed is demonstrated by a selec
tion of advanced applications in section 5, some of which are new as well (see also
[18, 22] for references to applications in economics, such as [19, 21]). To the interested
reader we also recommend [44, 48, 49, 56, 57, 59] for further background material and
orientations towards various applications in applied analysis and optimal control.
2 Narrow convergence of probability measures
This section recapitulates some results on narrow convergence of probability measures
on a metric space; cf. [2, 27, 28, 35, 46]. Let 5 be a completely regular Suslin space,
whose topology is denoted by r. On such a space there exists a metric p whose topology
TP is not stronger than r, with the property that the Borel cr-algebras B(S,TP) and
B(S,T) coincide. To see this, recall that in a completely regular space the points are
separated by the collection Cb(S^r) of all bounded continuous functions on 5. Since
S is also Suslin, it follows by [32, III.32] that there exists a countable subset (c¿) of
Cb(S, r), with sup^s |ct(^)| = 1 for each ¿, that still separates the points of 5. A metric
as desired is then given by p(x,y) := £ S i 2~l
ci(x) — C{(y). This is because rp C r
is obvious and by another well-known property of Suslin spaces, the Borel cr-algebras
B(S,p) and B(S,T) coincide [51, Corollary 2, p. 101]. Of course, if 5 is a metrizable
Suslin space to begin with, then for p one can simply take any metric on S that is
compatible with r.
As a consequence of the above, we shall write from now on
for respectively the Borel cr-algebra and the set of all probability measures on (5, B(S)).
Definition 2.1 (narrow convergence in V(S)) A sequence (vn) in V(S) converges
Tp-narrowly to u0 £ V(S) (notation: vn => v>0) if limn fs c dvn = fs c di/0 for every c in
Cb{S,rp).
Here Cb(S,Tp) stands for the set of all bounded rp-continuous functions from S into R.
Although Tp-narrow convergence is more fundamental for our purposes, we shall often
be able to use the stronger form of narrow convergence that arises when Cb(S, rp) in the
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![above definition is replaced by the larger set Cb(S,r). This will be denoted by " =
S
> ".
Definition 2.1 obviously extends to a definition of the r- and rp-narrow topologies on
V(S); we indicate these by TT and Tp). By [51, Appendix, Theorem 7] V(S) is a Suslin
space for TT (it is also Suslin and even metrizable for Tp - cf. [35, 111.60]). Hence,
completely analogous to what was observed above for 5, the Borel cr-algebras coincide
by [51, Corollary 2, p. 101]:
B(V(S)) := B(P(S),TT) = B(V(S),TP).
A vehicle by which we frequently manage to go from ^-convergence to the more general
r-convergence is r-tightness:
Definition 2.2 (tightness in V(S)) A sequence (i/n) in V(S) is said to be r-tight if
there exists a sequentially r-inf-compact function h : S —• [0, +oo] (i.e., all lower level
sets {x € S : h(x) < ft}, ft € R, are sequentially r-compact) such that supn fs h dvn <
-foo.
Observe that a fortiori h must be rp-inf-compact on S (causing h to be Borel measur
able); note here that rp is metrizable, so that the distinction between sequential and
ordinary rp-inf-compactness vanishes.
Remark 2.3 The above definition can be shown to be equivalent to the following one
[5, Example 2.5]: (Sn) is r-tight if and only if for every e > 0 there exists a sequentially
r-compact set Ke such that supn vn(SKe) < e.
Theorem 2.4 (portmanteau theorem for =>) (i) Let(i/n) and i/0 be inV(S). The
following are equivalent:
(a) vn £> i/0.
(6) limn Js c dvn — ¡s c di/0 for every c eCu(S,p).
(c) lim infn fs q dvn > fs q dv0 for every rp-lower semicontinuous function q : S —
»
(—oo,-foo] which is bounded from below.
(ii) Moreover, if (vn) is r-tight, then the above are also equivalent to
(d) i/n =£• I/Q.
(e) lim infn fs q di/n > fs q dvQ for every sequentially r-lower semicontinuous func
tion q : S —> (—oo,+oo] which is bounded from below.
Here Cu(S,p) stands for the set of all uniformly /o-continuous and bounded functions
from S into R. The name "portmanteau theorem" comes from [28].
Proof. Part (z), which is stated in a metrizable context, is classical; cf. [2, 4.5.1],
[27, Proposition 7.21] and [28, Theorem 2.1]. Next, we prove part (ii): (d) =* (a)
holds a fortiori, (e) => (d) is evident since for any c G Cb(S,r) both c and —c meet
the conditions imposed on q in part (e). (d) => (e): Let h be as in Definition 2.2
For any q as stated in (e) and for any e > 0 the function qt := q + eh is sequentially
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![r-inf-compact, whence rp-inf-compact. Hence, qt is rp-lower semicontinuous on S and
bounded from below.1
So (c) and an easy argument with e —• 0 give (e). QED
It turns out that tightness is a criterion for relative compactness in the narrow
topology. Just as in Definition 2.2 we only state the sequential version.
Theorem 2.5 (Prohorov's theorem for =>) Let (un) in V(S) be r-tight. Then
there exist a subsequence (vn>) of (i/n) and v* G V(S) such that vni =
S
> v+.
Proof. By T D TP we can apply Prohorov's classical theorem [28, Theorem 6.1]. Hence,
there exist a subsequence (vni) of (i/n) and v* G V(S) such that vn> => i/*. Hereupon,
we can invoke Theorem 2.4. QED
Let N := NU{oo} be the usual Alexandrov-compactification of the natural num
bers. This is a metrizable space, so let p be a fixed metric on N and let S := S x: N. We
can equip S with the product metric p or with the product topology f := r x r¿. For
n € N, let cn € V(N) be Dirac measure concentrated at the point n. The proof of the
next result is rather obvious by Theorem 2.4(6) and a triangle inequality argument.
Corollary 2.6 Let (i/n) and v>0 be in V(S). / / ^ E Î L i ^n =^ u0 in V{S), then
In particular, if vn => v0 in V(S), then
Recall that the support r-supp v of v G V(S) defined to be the complement of the
union of all open i/-null sets; hence, i/(r-supp v) = 1 (note that every r-open subset
of 5 has the countable subcover property by [35, 111.67]).
Theorem 2.7 (support theorem for =>) (i) Let ^E^=i v
n =^ ^o for (un) and u0
in V(S) (in particular, this holds when vn •=$> u0). Then
(ii) Moreover, if {vn) r-tight, then
i/0(T-seq-cl r-LsnT-supp vn) = 1 and r-supp u0 C r-cl T-Lsnr-supp vni
x
This shows q : S —
• R to be ^(5)-measurable, with q := q on {h < H-oo} and q := +oo on
{h = +00}. Hence, the integrals in (e) are well-defined.
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![Here "r-seq-cl" stands for sequential closure with respect to the topology r and "r-Lsn"
refers to the usual Kuratowski sequential r-limes superior of a sequence of subsets.
This set is r-closed if r = rp (metrizable case).
Proof, (i) By Corollary 2.6 it follows that TTN := jjYln=i{u
n x tn) 4» u0 x e^ in
V(S), where S := S x N. Setting Sn := rp-supp vn and Soo := rp-Lsnrp-supp vn, we
define q0 : S -» {0,-foo} as follows: If x G Sk then q0(x,k) := 0. If x £ Sk for all
k, 1 < k < oo, then ^o(^? &) := +oo. We claim that q0 is r^-lower semicontinuous in
every point (x,k) of S x N. For let /5((arJ
',fcJ
'),(a;, A:)) —
» 0. We must show that a :=
liminfn qo(xj
, P ) > q'0{x, k). If k < oo, then eventually F = A;, so a > qo(x, k) follows
since S*; is rp-closed. If k = oo, we can have two cases: if eventually kJ
' = oo, then a >
(jo(x, oo) follows by rp-closedness of S^. On the other hand, if F < oo infinitely often,
then the same inequality follows directly from the definition of Soo- This shows that q0
is indeed Tp-lower semicontinuous. Now fg qod(vn x en) = Js <jo(x, n)i/n(dx) = 0 for every
n. Hence, fsqodirjv = 0 for every N. Thus, Theorem 2.4 gives fsq0(x, oo)u0(dx) = 0,
and the desired rp-support property for VQ follows.
(iï) Since N is compact, f-tightness of (vn x en) in V(S) is evident. Hence, Theo
rem 2.4 gives jj ^nzzii^n x en) 4> v0 x e^ in V(S). We now essentially proceed as in
the proof of (¿), but a little more carefully: the additional sequential closure operation
in the definition of S^) is needed because r-Lsnr-supp vn need not be sequentially
r-closed on its own accord. QED
Theorem 2.8 Lei vn =$> v0 in V(S). Then (vn) is rp-tight.
Proof. 5 is Suslin, so any probability measure in V(S) is a Radon measure for both
r and rp [35, III.69]. Hence, the result follows from [28, Theorem 8, Appendix III].
QED
The above sufficient condition for rp-tightness of a sequence will play a role further
on. It seems to have no analogue for r-tightness when r is nonmetrizable. The
following result, also to be used later, is [27, Proposition 7.19]:
Proposition 2.9 (countable determination of =>) There exists a countable set Co
C { c G Cu(5, p) : sup5 |c| = 1} such that for every (i/n) and UQ in V(S) one has vn => i/0
if and only if lirrin fsc dvn = / 5 c dv0 for every c G Co. In particular, Co separates the
points ofV(S).
K-convergence of Young measures
A Young measure is a function 8 : il —
> V(S) that is measurable with respect to A
and B(V(S)) The set of all such Young measures is denoted by K{tt; S). By B(S) =
B(S)TP) of the previous section it is not hard to see that Young measures are precisely
the transition probabilities from (ÎÎ,-4) i n t o
(S,B(S)) [45, III.2], i.e., 6 : il -> V(S)
belongs to K(Q; S) if and only if u> i-» S(u;)(B) is .4-measurable for every B € B(S).
3
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![For some elementary measure-theoretical properties of Young measures the reader
is referred to [45, III.2] or [2, 2.6]. In particular, the product measure induced on
{Six S, Ax B(S)) by p and any S G ft(ft; S) (cf. [45, III.2]) is denoted by fi ® 6. Let
£°(iî;S) be the set of all measurable functions from (il, A) into (5, #(5)). A Young
measure S G %(il S) is said to be Dirac if it is a degenerate transition probability [45,
III.2], i.e., if there exists a function / € £°(il; S) such that for every UJ in Ü
Conversely, S is also called the Young measure relaxation of / . In this special case 6
is denoted by S = ej. The set of all Dirac Young measures in 7£(í); S) is denoted by
^Dirac(^'i S).
The fundamental idea behind Young measure theory is that, in some sense, 1Z(Ü; S)
forms a completion of £°(i); 5), when the latter is identified with 1Zoirac(^'i S).
Let us agree to the following terminology: an integrand on Í) x S is a function
^ : i î x S - > (—oo,+oo] such that for every w E Î Î the function g(u>, •) on S is B(S)-
measurable. A function g : il x S —
> (—oo, +oo] is said to be a (sequentially) r-lower
semicontinuous [r-continuous] [[r-inf-compact]] integrand on Í) x S if for every wGÎÎ
the function g(w, •) on S is (sequentially) T-lower semicontinuous [r-continuous] [[r-
inf-compact]] respectively. Let g be an integrand on fl x S. The following expression
is meaningful for any 6 G 7£(íl; S):
provided that the two integral signs are interpreted as follows: (1) for every fixed u the
integral over the set S of the function g(w, •), which is ¿?(5)-measurable by definition
of the term integrand, is a quasi-integral in the sense of [45, p. 41], (2) the integral
over Í7 is interpreted as an outer integral (note that outer integration comes down to
quasi-integration when measurable functions are involved - cf. [9, Appendix A] or [22,
Appendix B]). The resulting functional Ig : fc(il; S) —> [—oo, +oo] is called the Young
measure integral functional associated to g. Another integral functional associated to
g, this time on the set £°(i); S) of all measurable functions from Í1 into 5, is given by
the formula
The following notion of convergence was introduced and studied in a more abstract
context in [12, 13].
Definition 3.1 (K -convergence in 7£(S1; S*)) A sequence (£n) in 7£(iî; S) K -conver
ges with respect to the topology r to ¿>0 G 7£(iî; S) (notation: Sn —í- 60) if for every
subsequence (Sn/) of (6n)
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![Note that in the expression above the exceptional null set is allowed to vary with the
subsequence (6ni). We remark that À'-convergence is nontopological. If in the above
definition r is replaced by rp and " =
S
> " by " => ", we obtain the weaker notion of
K-convergence with respect to rp. This is denoted by " —^ ". We shall occasionally
use u
—> " in situations where we need not distinguish between the two at all.
Example 3.2 Let (íl,.4,¿0 be ([0,1],£([0,1]), Ai) (i.e., the Lebesgue unit interval).
Let (fn) be the sequence of Rademacher functions, defined by fn(u>) := sign sin(2n
7ru;)
(here S := R, of course). Then tjn —• 60l where S0 G 7£([0,1];R) is the constant
function S0((jj) = ~ti + |e_i. This can be proven by the (scalar) strong law of large
numbers, analogous to the proof of Theorem 3.8.
Definition 3.3 (tightness in K(il; S)) A sequence (Sn) in 7£(i); S) is r-tight if there
exists a nonnegative, sequentially r-inf-compact integrand h on il x S such that
supn Ih(Sn) < +oo.
This definition comes from [26]; clearly this extends Definition 2.2. Recall from the
previously given definition of integrands that a sequentially r-inf-compact integrand
h is simply a function on Q x S with the following property: for every w G Q the
function h(u),') is sequentially r-inf-compact.
Remark 3.4 Similar to Remark 2.3, Definition 3.3 can easily be shown to be equiva
lent to the following one [39]: (8n) is r -tight if and only if for every e > 0 there exists
a multifunction I : fî —
» 2s
, with TC(LU) sequentially r-compact for every u; G ÎÎ, such
that
Example 3.5 (a) Let E be a separable reflexive Banach space with norm || • ||.
Let E' be the dual space of E. Suppose that (/n) C ^(fyE) is £a
-bounded:
supn fu fn(uj)fi(du;) < +oo. Then (e/n) is a(E, Ef
)-tight in 7£(Í7;5): simply set
h(uû,x) := ||z|| in Definition 3.3.
(b) Let E be a separable Banach space with norm || • ||. Suppose that (/n) C
Cl
(Çl',E) is /^-bounded and that there exists a multifunction R : Í) —
> 2s
such that
for a.e. LJ both {/n(^) : n € N} C R(w) and R(u) is o~(E, ^^-ball-compact [i.e.,
the intersection of R(w) with every closed ball in E is cr(E, Ü^-compact]. Then (c/n)
is o~(E,E')-tigh.t: now we set hn(uj,x) := x if x G -#(<*;), and /i^(o;,x) := -foo
otherwise. Then for every u> € Q and ¡3 € R+ the set of all x G E such that hfi(uj, x) <
P is the intersection of R{OJ) and the closed ball with radius ¡3 around 0. By the
Eberlein-Smulian theorem it is sequentially cr(E, .¿^-compact as well.
Part (6) in the above example generalizes part (a): simply observe that in part (a)
E itself is a(E, E')-ball-compact by reflexivity, so there we can take R = E. A very
important property of A'-convergence of Young measures is as follows [13, 12, 18]:
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![Proposition 3.6 (Fatou-Vitali for —> ) (i) Let Sn —^ S0 in 1Z(Ü;S). Then
liminfn Ig(Sn) > Ig(So) for every rp-lower semicontinuous integrand g on H x S such
that
(3.1)
(ii) Moreover, if(Sn) is r-tight, then also liminfn Ig(Sn) > Ig{60) for every sequentially
T-lower semicontinuous integrand g on il x S such that (3.1) holds.
Here, as usual, g~ := max(—#,0) and {g < —a}^ denotes {x G 5 : g(u,x) < —a}.
Note that footnote 1 applies to each g(u;, •) in part (ii).
Remark 3.7 If Sn = c/n for all n G N, then (3.1) runs as follows:
Since g(íú,fn(uj)) < —a if and only if g~ (LÜ, fn(u>)) > a, (3.1) comes down to uniform
(outer) integrability of the sequence (g~(-, fn('))) in the case of a Dirac sequence, in
agreement with standard formulations; cf. [37, 5].
Proof of Proposition 3.6. The proof of (i) will be given in two steps.
Step 1: g > 0. Set fi := liminfn Ig(Sn); then there is a subsequence (Sn>) such
that P = Ymn. Ig(8n,). Define XI>N(U>) := }jT,n>=i Is 9(u,x)6n,(u;)(dx) and ^o(^) :=
Jsg((jü,x)ó0(u>)(dx). Then lim infjv I/>N > i¡)0 a.e. by Theorem 2.4(c), because by Defi
nition 3.1 jfYln'=i àn'{u) 4> S0(LÜ) in V(S) for a.e. w. Thus, Fatou's lemma can be
applied (it remains valid for outer integration in the direction that suits us; cf. [22,
Appendix B]). This gives /? > liminf;v^oo JQ ^vcfy/ > JQ ipodfi = Ig(S0) by subadditivity
of outer integration.
Step 2: general case. We essentially follow Ioffe [37] by pointing out that
J g(u,x)6n(u))(dx) + J l{g<-ct}(Lü,x)g-(uj,x)ón(u;)(dx) > J gQ(<jj,x)6n(uj)(dx),
by ^r -f- l{g<_Qyg~ > ga := max(#, —a). One more (outer) integration gives, in the
notation of (3.1), Ig(Sn) + s(a) > Iga(Sn), where we use subadditivity of outer integra
tion. Now step 1 trivially extends to any g that is bounded from below, such as ga.
This gives
liminf Ig(6n) + s(a) > liminf Iga(6n) > I9a(S0) > Ig(60),
where we use ga > g. The proof of (i) is finished by letting a go to infinity.
(¿¿) Let h be as in Definition 3.3 and denote s := supn Ih(6n). We augment g, similar
to the proof of Theorem 2A(ii): For e > 0 define gt
:= g--eh. Then gt
> g and #c
(u;, •)
is Tp-lower semicontinuous on S for every UJ G 0 (see the proof of Theorem 2.4(H)).
Thus, part (i) gives liminfn Ig(Sn) + es > liminfn Igc(6n) > Ig<(60) > Ig(60) for any
e > 0. Letting e go to zero gives the desired inequality. QED
The following important Prohorov-type relative compactness criterion for /^-conver
gence is [13, Theorem 5.1]. It was obtained as a specialization to Young measures of
an abstract version of KomlóV theorem [41]; see also [14].
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![Theorem 3.8 (Prohorov's theorem for - ^ ) Let (Sn) in ft(ft; S) be r-tight
A' T
Then there exist a subsequence (6n>) of (Sn) and Sm G 1Z(Çl; S) such that 6n> —^ 6*.
To prove Theorem 3.8 we use the following theorem, due to Komlós [41].
Theorem 3.9 (Komlós) Let (0n) be a sequence in Cx
(ft; R) such that supn fQ (/>nd/ji
< +oo. Then there exist a subsequence ((¡>nf
) of (<f>n) and a function </>+ G £1
(Í7;R)
such that for every further subsequence (<j)n») of (<j>n')
Lemma 3.10 Let (un) in V(S) be T-tight and let Co be a subset of {c G Cb(S,r) :
sup5 c = 1} that separates the points ofV(S). If
then there exists v* G V{S) such that vn =
S
> i/*.
This lemma is a direct result of Theorem 2.5 and the point separating property of C0;
cf. Proposition 2.9.
Proof of Theorem 3.8. Let Co = {c, : i G N} be as in Lemma 3.10. Define
<f>i,n(u) := fs Ci(x)6n(u>)(dx); then supn fQ |0¿,n|c?// < +°o for every i G N. Let h be
as in Definition 3.3. By definition of outer integration, there exists for each n G N a
function (j>o,n G £l
(Q;R) such that (j>o,n{w) > fsh(uj,x)8n{uj)(dx) for a.e. UJ G O and
In 0o,n dfx = Ih(8n). Applying Theorem 3.9 in a diagonal extraction procedure, we
obtain a subsequence (Snt) of (8n) and functions <t>^m G £1
(Í);R), i G N U {0}, such
that liiriiv ^ Z¡^/=i <^î,n" = <t>i,* a.e. for every further subsequence (6n») and for all
i G N U {0}. Explicitly, we have every such (Snn) for a.e. u in Q
(3.2)
(3.3)
Let us first consider (Sn>) itself as the subsequence in question. Fix to outside the
exceptional null set M, associated with this particular choice of a subsequence in
(3.2)-(3.3). Then (3.2) implies that the sequence (vN) in V(S), defined by vN :=
jjY^n'=iàn'(u)i is r-tight in the classical sense of Definition 2.2. Also, (3.3) implies
that im.N ¡s Cidvjv exists for every i. By Lemma 3.10 there exists v^,* in V(S) such
that z//v 4> UUJ^. Define ¿>*(u>) := i^u;,* for u G ÜM. Also, on M we define 6+ to be
equal to an arbitrary fixed element of V(S). Then it is elementary to show that Sm
32](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-45-320.jpg)
![4
belongs to 7£(iï;5). The argument following (3.3) can be repeated with a change of
the null set M (for which Definition 3.1 leaves room) if one starts out with an arbitrary
subsequence (6nn) of (6nt). QED
The next example extends Example 3.2 and demonstrates the power of Theo
rem 3.8, which brings K-convergence (for subsequences!) to settings where Kol-
mogorov's law of large numbers, used in the special Example 3.2, stands no chance at
all.
Example 3.11 Let (iî,,4, //) be ([0,1],£([0, l]),Ai) (i.e., the Lebesgue unit interval).
Let /i € >C1
([0,1]; R) be arbitrary; it can be extended periodically from [0,1] to all of
R. We define /n+i(u;) := /i(2n
u;). Clearly, the sequence (e/n) is tight in the sense of
Definition 3.3 (see Example 3.5(a)). By Theorem 3.8 there exist a subsequence (/n/)
of (/n ) and some 6* £ 7£([0,1]; R) such that e/n, —• £*. The precise nature of 6m can
now be determined by means of Proposition 3.6, but we shall defer this to Example 4.4
later on.
The following are direct consequences of Corollary 2.6 and Theorem 2.7 for K-
convergence of Young measures (by their application pointwise):
Corollary 3.12 Let (Sn) and S0 be in 7£(fi;S). The following are equivalent:
(a) 6n —^ ¿o in H(T; S)
(b) 6n x en ^ So x Coo in H(T;S).
Theorem 3.13 (support theorem for —•» ) (i) Let Sn —^ S0 in 1Z(Q;S). Then
Tp-supp 60(u>) C Tp-LsnTp-supp Sn(uj) for a.e. UJ inVt.
(ii) Moreover, if Sn —U 60, then also
S0(u;)(r-seq-c r-Lsnr-supp 6n(w)) = 1,
r-supp ¿o(k>) C r-cl r-Lsnr-supp Sn(uj) for a.e. u in 0.
Narrow convergence of Young measures
In this section our program to transfer narrow convergence results for probability mea
sures (section 2) to Young measures comes is completed. We use the same fundamental
hypotheses as in the previous section: (fi, .4, fi) is a finite measure space and (5, r) is
a completely regular Suslin space, on which we also consider the weak metric topol
ogy rp C T. We start out by giving the definition of narrow convergence for Young
measures [3, 4, 10] (see also [38]).
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![Definition 4.1 (narrow convergence in 7l(T] S)) A sequence (Sn) in 1Z(Ü; S) con
verges r-narrowly to S0 in 7£(fi; S) (this is denoted by Sn =^> S0) if for every A G A
and c in C&(5, r)
The weaker notion of rp-narrow convergence is defined by replacing r by rp this is
denoted by " = > ". We shall occasionally use " = > " in situations where we need
not distinguish between the two at all. We shall see that on r-tight sets of Young
measures these two modes actually coincide (note the complete analogy to section 2).
For further benefit, note carefully the distinct notation used for narrow convergence
for probability measures (indicated by short arrows) and Young measure convergence
(indicated by long arrows).
Remark 4.2 ( —-+ implies = > ) Let (Sn) and So be in 7£(fi; S). The following hold:
(a) If Sn —^ S0, then Sn = > So. (6) If Sn —^ So and if(Sn) is r-tight, then Sn =^=> So.
(c) If Sn —^ S0, then Sn =?=> S0.
Definition 4.1 obviously extends to a definition of the r- and rp-narrow topologies.
In the form given above, the definition of narrow convergence is classical in statis
tical decision theory [58, 43]. It merges two completely different classical modes of
convergence:
Remark 4.3 Let (Sn) and S0 be in 7£(ft;5). The following are obviously equivalent:
(a) Sn =^> ¿o in 1Z(ft; S).
(b) [fi®6n](A x -)/fi{A) ^ [fi®6o](A x -)/fi(A) in V(S) for every A G A, fx(A) > 0.
(c) fs c(x)Sn(-)(dx) -^ fsc(x)S0(')(dx) in £°°(ft;R) for every c G Cb{S,r). Here
"—>> " denotes convergence in the topology a(C°°(ft; R),£a
(iî; R)).
The following example continues the previous Examples 3.2 and 3.11.
Example 4.4 Let (il,A,fi) be ([0,1],£([0,1]), Ai) (cf. Example 3.2). As in Exam
ple 3.11, let /i G >C1
([0,1]; R) be arbitrary and extended periodically from [0,1] to all
of R. We define / n + i M := fi(2n
u>). Then efn = * 60, where S0 G ft([0,l],R) is the
constant function given by S0(uj) = Af1
. Here A* G V(H) is the image of Ai under the
mapping / i ; i.e., X^(B) := A(/1 ~1
(JB)). TO prove the above convergence statement,
let c G C&(R) be arbitrary, and let A be first of the form A = [0,/?] with /? > 0.
Then a simple change of variable gives limn^oo fA c(fn)di = fA[fRc(x)So(u))(dx)]duj
for A = [0,/?]. By standard methods this can then be extended to all A in A.
It follows that Sm in Example 3.11 is equal to the above ¿0, modulo a Ai-null set. The
proviso of an exceptional null-set is indispensible, because the narrow limits in 7£(ft; S)
are only unique modulo a /¿-null set:
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![Proposition 4.5 For every 6, 8' in 71(0,] S) the following are equivalent:
(a) ¡A[fsc(x)6(uj)(dx)}fi(duj) = ¡A[JS c(x)ó'(uj)(dx)}fi(duj) for every A £ A and
ce C0.
(b) 6(u>) = 6'(UJ) for a.e. u> in 0,.
Theorem 4.6 Suppose that the a-algebra A on O ¿5 countably generated. Then there
exists a semimetric dn on 7£(il; S) such that for every (Sn) and S0 in 1Z(íl; S) the
following are equivalent:
(a) 6n => S0.
(b) imndn(Sn,60) = 0.
Proof. Let (c¿) enumerate Co of Proposition 2.9, and let (Aj) be the at most countable
algebra generating A. Denote qi(A,S) := fA[fsc
iô(')(dx
))dv an(
^ define a semimetric
on ft(il; S) by dn(6,6') := Ei,j 2"^'|ft(A¿, S) - qi(Ah 8'). To prove (a) =» (6) we note
that standard arguments [2, 1.3.11] give limn qi(A, 8n) = qi(A, S0) for every A € A and
i. By Proposition 2.9 and Remark 4.3 this implies 6n = > S0. Conversely, (a) =ï (b) is
simple. QED
Proposition 3.6 and Theorem 3.8 imply the following transfer of the earlier port
manteau Theorem 2.4 to the domain of Young measures [10].
T h e o r e m 4.7 (portmanteau t h e o r e m for =>) (i) Let (Sn) and 80 be in 7£(f&; S).
The following are equivalent:
(a) 6n =£> 60.
(6) mn ¡A[fs c(x)6n(u;)(dx)]p(duj) = ¡A[fs c(x)S0(iü)(dx)]p(duj) for every A € A,
ceCu(S,p).
(c) liminfn Ig(Sn) > Ig(So) for every rp-lower semicontinuous integrand g on il x S
such that
(ii) Moreover, if (6n) is T-tight, then the above are also equivalent to
(d) Sn =^> S0.
(e) liminfn Ig{àn) > Ig(õo) for every sequentially r-lower semicontinuous integrand
g on Q x S such that
Proof. By Remark 4.3 (a) & (b) follows by (a) <
£
> (6) in Theorem 2.4. (c) => (6)
is obvious: apply (c) to g(w,x) := ±lA(u;)c(x). (a) => (c): By Remark 4.3 vn 4> v0,
where vn := [p (g) 6n](Çl x -)/p(il). So by Theorem 2.8 (vn) is rp-tight in V(S): there
exists a Tp-inf-compact h' : S —• [0, + oo] such that supn fs h!dvn < +oo. So (Sn) is
rp-tight, since fsh'di/n = Ih(Sn)/p(0,) for h(w,x) := hx), Therefore, Theorem 3.8
applies to (Sn). For g as stated, let ¡3 := liminfn Ig(Sn). Then ¡3 = limn/ Ig(ón>) for
a suitable subsequence (6n>) and, by Theorem 3.8, we may suppose without loss of
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![generality that Sn> —^ 6+ for some 6* in 11(0,; S). But in combination with (a) this
implies ¿*(u>) = S0(LO) a.e. (Proposition 4.5), so in fact 8n> —^ ¿0. Now ¡3 > Ig(S0)
follows from Proposition 3.6. Next, (d) => (a) holds a fortiori and (a) => (e) is proven
similarly to (a) => (c), but now r-tightness holds ab initio; let h be as in Definition 3.3.
In the remainder of the proof of (a) => (c) we now substitute ge
:= g + eh, which is
a Tp-lower semicontinuous integrand. Letting e —
» 0 gives (e). Finally, (e) => (c?) is
obvious. QED
Results of this kind (but less general) are usually obtained by means of approxima
tion procedures for the lower semicontinuous integrands [31, 26, 3, 38, 5, 10, 56, 57],
that are completely avoided here. Another difference is that the present approach
directly produces results for sequential Young measure convergence.
A'
Theorem 4.8 (characterization of = > by —> ) (i) Let(Sn) andó0 be inK(0;S).
The following are equivalent:
(a) Sn = > S0.
(b) Every subsequence (¿n>) of (6n) contains a further subsequence (6n») such that
6n» —^ S0.
(ii) Moreover, if (Sn) is r-tight, then the above are also equivalent to
(c) Sn =h> S0.
(d) Every subsequence (6n>) of (Sn) contains a further subsequence (8n») such that
Sn» —^ ¿o-
In parts (6) and (d) the use of subsequences cannot be replaced by the use of the entire
sequence (Sn) itself, simply because a narrowly convergent sequence does not have to
A'-converge as a whole [18, Example 3.17].
Corollary 4.9 (i) Let (Sn) and S0 be in 1Z(Q;S). The following are equivalent:
(a) Sn => S0 in 1Z(il;S).
(b) Sn x en =£» So x Coo in 11(0,; S).
(ii) Moreover, if (Sn) is r-tight, then the above are also equivalent to
(c) Sn =^ S0 in 11(0; S).
(d) Sn x en =^> S0 x Coo in 11(0; 5).
Proof, (a) <
=
> (6) is immediate by Theorem 4.8 and Corollary 3.12. (a) <& (c) is
contained in Theorems 4.7 and 4.8. (b) <& (d) is contained in Theorems 4.7 and 4.8,
since (Sn x en) is f-tight if and only if (Sn) is r-tight (by compactness of N). QED
Transfers of Prohorov's theorem and of the support theorem to Young measure
convergence are immediate because of the intermediate results obtained in section 3.
The following result is evident by combining Theorem 3.8 and Remark 4.2. See [11]
for the topological (i.e., nonsequential) version of this result in precisely the setting of
this paper.
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![Theorem 4.10 (Prohorov's theorem for = > ) (i) Let (8n) in TZ(il;S) be rp-tight.
Then there exist a subsequence (6ni) of (8n) and S* G 7£(í); S) such that 8n> = > 8*.
(ii) Let (8n) in 1l(iï;S) be r-tight. Then there exist a subsequence (8n>) of (8n) and
8* G 7£(íl; S) such that 8n> =^=> 8*.
Example 4.11 We continue with Example 3.5(6). By a(£, E')-tightness of (e/n) we
get from Theorem 4.10 that there exist a subsequence (/n/) of (fn) and 6m G 7^(0; E)
such that tf , => 8*.
(a) We now introduce a function /* G Cl
E that is "barycentrically" associated to ¿*,
simply by inspecting the consequences of the tightness inequality s :— supn hR{tfn) <
+oo that was established there. For hft is a fortiori a cr(E, £")-lower semicontinuous in
tegrand, so Theorem 4.7(e) gives hR(8*) < s < H-oo, which implies fs /&/?(u;, x)8*(uj)(dx)
< +oo for a.e. UJ. So by the definition of hp. it follows that both 8m(u;)(R(uj)) = 1 and
SE x8*{w)(dx) < H-oo for a.e. UJ. By standard facts of Bochner integration it follows
that the barycenter /*(u;) := bar ¿*(u;) of the probability measure 8*(ÙJ) is defined
for a.e. UJ. Thus, if we set /* := 0 on the exceptional null set, we obtain a func
tion /* G £°(íl;E). Finally we notice that, as announced, /* is /i-integrable, i.e.,
/* G Cl
(£l',E). This follows simply from hR(8*) < +oo by use of Jensen's inequality
and the inequality hn(uj,x) > x.
(6) Suppose that in part (a) one has in addition that (||/n'||) is uniformly integrable
in Cl
(£l;R). Then fn> A /* G Cl
(Q;E) (weak convergence in £1
(Í2;E)). This follows
directly from another application of Theorem 4.7(e), namely, to all integrands g of
the type g(uj,x) = ± < x,b(uj) >, b G C°°(Q, E')[E}. The latter symbol denotes
the set of all scalarly measurable bounded E'-valued functions on Í); it forms the
prequotient dual of £1
(fî; E). This yields limn/ Ig(tfn,) = Ig(8*), with Ig(cfn,) = Jg(fn')
= in < fw,b(u) > dfi and Ig(6m) = ¡Q < f*,b(u>) > dp.
Part (6) in the above example implies that fn A f0 in Example 4.4, where /0 is the
constant function given by fo(u>) := bar 8O(UJ) = JnfidXi (apply [35, 11.12]). Concate
nation of Theorem 3.13 and Theorem 4.8 gives immediately the following result:
Theorem 4.12 (support theorem for = > ) (i) Let 8n = ^ 80 in 7£(i);£). Then
Tp-supp 8O(LJ) C rp-Lsnr/7-supp 8n(uj) for a.e. UJ in il.
(ii) Moreover, if (8n) is r-tight, then 8n =^> 80 in 7£(f); S) and
¿o(u;)(T-seq-cl r-Lsnr-supp 8n(uj)) = 1,
r-supp 8Q(UJ) C r-cl r-Lsnr-supp 8n(uj) for a.e. UJ in il.
The following so-called lower closure theorem for Young measures forms a combi
nation of the main relative compactness, lower semicontinuity and support results of
the present section. Let (D,do) be a metric space.
37](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-50-320.jpg)
![Theorem 4.13 (fundamental lower closure theorem) Let (6n) in K(iï;S) be r-
tight and let dn —• do in C°(Q,; D) (convergence in measure). Then there exist a sub
sequence (Snt) of (Sn) and 6* in 7£(0; S) such that
for every sequentially f -sequentially lower semicontinuous integrand £ on Q x S x D)
such that
(4.1)
More precisely, we have causing 6* to be supported as follows
Here {£ < —a}w > n stands for the set of all x G S for which £(UJ, x, dn(u>)) < —a.
Proof. Theorem 3.8 and well-known facts about convergence in measure [28, The
orem 20.5] imply existence of a subsequence (6n',dn>) of (6n,dn) and existence of
6* G 7Z(T;S) such that 6n> —^ 6+ in 7£(í); S) and dr)(dnt(u>),d0(uj)) —> 0 for a.e.
LO. A fortiori this gives 8n> =^=> ¿* (by Remark 4.2). By Theorem 4.12 this gives the
desired pointwise support property for 6*. By Corollary 4.9, we also have 6nt =^> 6*
in 1Z(ft; S), with Sn := 6n x en and 6* := ó* x e^ Without loss of generality we discard
renumbering and pretend that (nf
) enumerates all the numbers in N. For £ as stated
we observe that for each n' G N the following identity holds
and it continues to hold for n' — oo if we write d^ := d0 and 6^ := 6*. Here
gt(w, (x,k)) := £(tu,x,dk(w)) defines a f-lower semicontinuous integrand g¿ on Q, x S
(modulo an insignificant null set). Note in particular that for k = oo lower semicon-
tinuity of ge(w, •) at (#,oo) follows from dn>(uj) —> d0(u>) and lower semicontinuity of
¿(u;,-,-) at (x,do(üj)). Thus, the desired inequality is contained in liminfn/ Igt(8ni) >
Igt(8*), a result that follows by applying Theorem 4.7 to g¿ (observe here that (4.1)
coincides with (3.1) for g = ge). QED
Remark 4.14 Let h be the nonnegative, sequentially r-inf-compact integrand h on
ClxS that corresponds as in Definition 3.3 to the r-tight sequence (Sn) in Theorem4-13;
i.e., with s := supn Ih(ôn) < +oo. Then the uniform integrability condition (4-1)
applies whenever the integrand £ has the following growth property with respect to h:
for every e > 0 there exists <¡>e G £X
(H; R) such that for every n G N
£~(uj,x,dn(uj)) < ch(w,x) + <¡>t(u) on O x xS.
Indeed, we can observe that the set {£ < —a}w,n in (4-1) is contained in the union
of {(j)e < eh} and {(¡>e > a/2}, which gives s'(a) < 3es + f{4>e>a/2} fa ^Pi whence
sf
(a) —* 0 for a —» oo, as claimed.
38](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-51-320.jpg)
![5 Some applications to lower closure and dense-
ness
We illustrate the power of the above apparatus by some applications to a variety of
problems; we refer to [18, 22] for more extensive expositions.
As our first application, we derive an extension of the so-called fundamental theo
rem for Young measures in [25]. Here L is a locally compact space that is countable at
infinity; its usual Alexandrov compactification is denoted by L := LU{oo}. The space
L is metrizable, and its metric is denoted by d. On L we use the natural restriction
of </, and denote it by d. Let CQ(L) be the usual space of continuous functions on
L that converge to zero at infinity. Although it could be avoided by the additional
introduction of transition subprobabilities (see the comments below), the Alexandrov
compactification L of L figures explicitly in the result. Also, below v denotes a cr-finite
measure on (Í!, .4).
Corollary 5.1 (i) Let (fn) in £°(il;L) and the closed set C C L be such that limn
v(f~x
(LG)) = 0 for every open G, C C G C L. Then there exist a subsequence (fn>)
°f (In) and 6* in 1l(Q; L) such that 6*(u>)(LC) = 0 for a.e. u in Í1 and
for every <j> G £a
(í2; R) and every c G Co(L).
(ii) Moreover, if for that subsequence there exists a sequence (Kr) of compact sets in
L such that limr_>oo supn/ U({LO G fi : fn'{u) $ Kr} = 0 then ¿*(u;)({oo}) = 0 for a.e.
u in Í7 and
for every A G A, <> G CX
(A', R) and c G C(L) for which (lJ4c(/n/)) is relatively weakly
compact in £1
(i4;R).
In [25] both L and H are Euclidean, and the AVs are closed balls around the origin with
radius r. As was done in [25], the result could be equivalently restated in terms of the
transition swôprobability 6'm from (i),^4) into (!/,#(!/)), defined by obvious restriction
to Z/, i.e., SI(UJ)(B) := 6*(u>)(B U {oo}), B G B(L). In this connection the tightness
condition in part (ii) guarantees that 6* is an authentic transition probability (Young
measure). Rather than via (¿), part (ii) could also have been derived directly from
Theorem 3.8 or 4.13.
Proof, (i) By cr-finiteness of z/, there exists a finite measure fi that is equivalent
to v. Let ^ be a version of the Radon-Nikodym density dis/dp. Now (¿>n), defined
by Sn := tfn G 7£(fi, ;¿), is trivially tight by compactness of L (set h = 0). By
Theorem 3.8 or 4.13 (with S := L, p := c?), there exist a subsequence (fn>) of (/n)
and 6* in H(Q; L) for which e/n, = > 6* (and even tfn, —^ 6*). Every c G C0(L) has
39](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-52-320.jpg)
![a canonical extension c G Cb(S) by setting c(oo) = 0. Now (¡><j) is //-integrable for any
</> G Cl
(Çl,A, J/;R), and Theorem 4.7(c) (or 4.13) can be applied to g : il x L —
> R
given by g(u,x) := ±</>(L>)<¡>(U)C(X). This gives the desired equality, because of the
identity ¡Q ¿(J) fL c(x)6*(-)(dx)dfi = ¡Q<f>fLc(x)6*(-)(dx)di/.
Next, let C be as stated. For any i G N the set F¿, consisting of all x G L
with ¿/-dist(x, C) < ¿- 1
, is closed in Z. Note already that nt F, = C, by the given
Td-closedness of C in L. Further, F{ := F{ U {00} is closed in L. Set gi(u>,x) :=
<^>(o;)l5vp,.(a;). This defines a nonnegative lower semicontinuous integrand p, on il x L.
Hence, 4(¿„) < # := liminfn, 4(£/ n ,) by Theorem 4.7(c). By SFi = LF¡, the
definitions of g{ and e/n, give 4(e/ n ,) = i/(/-/(IFt )). So # = liminfn/ */(/-/(LF¿))
< Í/(/~/1
(XG¿)), where G¿, Gt C Ft , is the r^-open set of all x G L with c?-dist(x, C) <
i~l
. Since G¿ D C, the hypotheses imply 0 = /?,- > Igx{à*) = /nÃ*(-)(LF,-)di/. Hence
6*(u)(LC) = 0 z/-a.e. because of fltF¿ = C, which was demonstrated above.
(¿z) The additional condition is a tightness condition for (c/n), when viewed as a
subset of 7Z(Q; L) (take I = Kr in Remark 3.4). Hence, there is a rp-inf-compact
integrand h on Q x L with supn Ih(ôn) < +00. Now define the inf-compact integrand
h on Q x L by h(u;,x) := h(u>,x) if x G X and Ã(u;,oo) := +00. Then /¿(Ó*) <
liminfn//^(e/n/) < +00 by Theorem 4.7(c). Hence, £*(-)({°°}) = 0 /¿-a.e., whence
z/-a.e. Finally, for any A G -4 with ^(A) < +00 Theorem 4.7(c) applies to g(w,x) :=
±IA(W)<I>(W)<J>(W)C(X). This gives the desired limit statement. If v(A) = +00 and A is
as stated, there exists a sequence (Aj) of subsets of A with finite i/-measure, with Aj |
A. The previous result applies to each of the Aj and the weak relative compactness
hypothesis implies uniform <r-additivity [30], so also in this case the desired limit
statement follows. QED
Next, let E and F be separable Banach spaces, each equipped with a locally convex
Hausdorff topology, respectively denoted by TE and r/r, that is not weaker than the
weak topology and not stronger than the norm topology. Let (D^do) be a metric
space. Functions that are "barycentrically" associated to Young measures can play a
special role in lower closure and existence results. This is demonstrated by our proof
of the following result.
Theorem 5.2 Letdn —> d0 inC°(il]D) (convergence in measure), en A eo ¿n£1
(0;£')
(weak convergence), and let (fn) in £1
(i);F) satisfy supn fQ fnFdfi < +00. Suppose
that there exist TE- and TF-ball-compact multifunctions RE ' ft —
> 2E
and Rp : Q, —• 2F
such that
{(en(u))Jn(u>)) : n G N} C RE{u>) x RF{u)/i-&.e.
Then there exist a subsequence (dn>, en/,/n/) of(dn,enyfn) and f* G £1
(Í2;F) such that
for every sequentially TEXTFX Try-lower semicontinuous integrand Í on fix (Ex Fx D)
such that the following hold:
40](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-53-320.jpg)
![£{w, -, -,do(u>)) is convex on E x F for a.e. UJ.
Moreover, the functions e$ and /* can be localized as follows: 2
(co(c«;),/*(a;)) G cl co-w-Lsn{(en(u;),/n(u;))} for a.e. u> in ÍL
Observe, as was already done following Example 3.5, that the ball-compactness condi
tion involving RE and Rp is automatically satisfied in case the Banach spaces E and
F are reflexive.
Proof. To apply Theorem 4.13 we set S := EXF,T := TEXTF and Sn := C(Cn,/n). Then
S is completely regular (by the Hahn-Banach theorem) and Suslin. Note that (||en||)
in £a
(fi; R) is uniformly integrable by [30, Theorem 1] and [45, Proposition II.5.2]. In
particular, this implies supn Jn ||(en ,/n )||s dfi < +oo. This proves that (Sn) is r-tight,
in view of Example 3.5(6). We can now apply Theorem 4.13: let the subsequence
(Sn>,dni) of (6n,dn) and 6* in 7£(í); S) be as guaranteed by that theorem, i.e., with
6ni = ^ 6+ (and even 6n> —^ 6*). Then it is elementary by Definition 4.1 that, U
E-
marginally", e6n/ ^ ¿f and, "F-marginally", efn, =^> if. Hereof (a;) := i,(w)(.xf),
etc. Then ^-marginally Example 4.11(6) applies, which gives that bar Sf = e0 a.e.
Also, F-marginally Example 4.11(a) applies, giving the existence of /* G £1
(Q; F) such
that /* = bar i f a.e. (note that r#- and rp -ball-compactness imply a(E^E')- and
cr(F, F')-ball-compactness respectively). Recombining the above two marginal cases,
we find bar 6+ = (e0,/*) a.e. (note that barycenters decompose marginally).
We now finish the proof. For an integrand £ of the stated variety Theorem 4.13
gives
where ¡3 := liminfn/ /¿¡^(u;,en/(o;),/n/(a;),dn/(a;))//(du;). In the inner integral of the
above inequality the convexity of £(UJ, -, -, d0(u>)) gives
for a.e. u?, by Jensen's inequality and our previous identity bar 6+ = (eo,/*) a.e. The
desired inequality thus follows. QED
The above lower closure result "with convexity" is quite general: it further extends
the results in [5, 8], which in turn already generalize several lower closure results in
the literature, including those for orientor fields (cf. [33]). See [15] for another de
velopment, not covered by the above result. Results of this kind are very useful in
the existence theory for optimal control and optimal growth theory; e.g., see [33, 15].
Recently, similar-spirited versions that employ quasi-convexity in the sense of Morrey
have been given in [42, 52] (these have for en the gradient function of dn and depend
on a characterization of so-called gradient Young measures [40, 48]). Corollaries of
2
In case E and F are finite-dimensional one may replace here "cl co" by "co".
41](https://image.slidesharecdn.com/19178557-250519113502-51422eef/85/Calculus-Of-Variations-And-Optimal-Control-1st-Edition-Ioffe-54-320.jpg)
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- 6. Calculus of variations and optimal control Technion 1998
- 7. CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne(Founding Editor) Editorial Board H. Amann, University ofZurich B. Lawson, State University of New York R. Aris, University ofMinnesota at Stony Brook G.I. Barenblatt, University of Cambridge B. Moodie, University of Alberta H. Begehr, Freie Universitàt Berlin S. Mori, Kyoto University P. Bullen, University of British Columbia L.E. Payne, Cornell University R.J. Elliott, University of Alberta D.B. Pearson, University ofHull R.P. Gilbert, University of Delaware I. Raeburn, University of Newcastle R. Glowinski, University ofHouston G.F. Roach, University ofStrathclyde D. Jerison, Massachusetts Institute of I. Stakgold, University of Delaware Technology W.A. Strauss, Brown University K. Kirchgàssner, Universitàt Stuttgart J. van der Hoek, University of Adelaide Submission of proposals for consideration Suggestions for publication, in the form of outlines and representative samples, are invited by the Editorial Board for assessment. Intending authors should approach one of the main editors or another member of the Editorial Board, citing the relevant AMS subject classifications. Alternatively, outlines may be sent directly to the publisher's offices. Refereeing is by members of the board and other mathematical authorities in the topic concerned, throughout the world. Preparation of accepted manuscripts On acceptance of a proposal, the publisher will supply full instructions for the preparation of manuscripts in a form suitable for direct photo-lithographic reproduction. Specially printed grid sheets can be provided. Word processor output, subject to the publisher's approval, is also acceptable. Illustrations should be prepared by the authors, ready for direct reproduction without further improvement. The use of hand-drawn symbols should be avoided wherever possible, in order to obtain maximum clarity of the text. The publisher will be pleased to give guidance necessary during the preparation of a typescript and will be happy to answer any queries. Important note In order to avoid later retyping, intending authors are strongly urged not to begin final preparation of a typescript before receiving the publisher's guidelines. In this way we hope to preserve the uniform appearance of the series. CRC Press UK Chapman & Hall/CRC Statistics and Mathematics Pocock House 235 Southwark Bridge Road London SEI 6LY Tel: 0171 407 7335
- 8. Alexander Ioffe Simeon Reich Itai Shafrir Israel Institute of Technology (Editors) Calculus of variations and optimal control Technion 1998
- 9. First published 2000 by Chapman & Hall Published 2020 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2000 by Taylor & Francis Group, LLC CRC Press is an imprint ofTaylor & Francis Group, an lnforma business No claim to original U.S. Government works ISBN-13: 978-1-58488-024-0 (pbk) ISBN-13: 978-1-138-40406-9 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences oftheir use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders ifpermission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system ofpayment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Library of Congress Cataloging-In-Publication Data Catalog record is available from the Library of Congress.
- 10. Preface to Volume II This is the second volume of the refereed proceedings of the International Conference on the Calculus of Variations and Related Topics held during 25-31 March, 1998 at the Technion - Israel Institute of Technology. This conference was sponsored by the Institute of Advanced Studies in Mathematics at the Technion with support from the United States National Science Foundation. The conference commemorated 300 years of the Calculus of Variations and brought together some of the leading experts in this very active field of research. In addition to the undersigned, the organizing committee of the conference also included Roger D. Nussbaum and Hector J. Sussmann of Rutgers University. The Calculus of Variations is a classical, yet of great current interest, area of Mathematical Analysis with manifold applications in Science and Technology. The papers assembled in these proceedings provide an overview of the state of the art in many branches of this important subject. All the papers were refereed by experts and revised whenever necessary. The papers in this second volume are mainly devoted to variational aspects of Optimal Control while the papers in the first one are mainly devoted to Critical Point Theory and to variational methods in Differential Equations. We are very grateful to the other members of the organizing committee for all their work, to the speakers and participants of the conference, to the contributors to these proceedings, and to the referees who kindly provided expert advice. We are pleased to acknowledge the generousfinancialsupport of the NSF. Thanks are also due to Haim Brezis for his help and advice with regard to the publication of these proceedings. We are indebted to Mary Lince of CRC Press for her patient cooperation throughout the publication process. Finally, we thank Sylvia Schur, the Secretary of the IASM, for her indispensable help in the organization of the conference and the preparation of these proceedings. Alexander Ioffe, Simeon Reich and Itai Shafrir Haifa, April 1999
- 12. Contents Preface Z Artstein Variational limits constrained by measure-valued multifunctions 1 E J Balder New fundamentals of Young measure convergence 24 A V Dmitruk and A A Milyutin A condition of Legendre type for optimal control problems, linear in the control 49 A L Dontchev and K Malanowski A characterization of Lipschitzian stability in optimal control 62 H O Fattorini Some remarks on the time optimal control problem in infinite dimension 77 H Frankowska and S Plaskacz Hamilton-Jacobi equations for infinite horizon control problems with state constraints 97 L Freddi Limits of control problems with weakly converging nonlocal input operators 117 V Kaminsky Heavy balls minimizing convex function in conditional average 141 A A Milyutin Calculus of variations and optimal control 159 B S Mordukhovich Existence theorems in nonconvex optimal control 173 N P Osmolovskii Second-order conditions for broken extremal 198 S Pickenhain and M Wagner Critical points in relaxed deposit problems 217 T Roubicek Convex locally compact extensions of Lebesgue spaces and their applications 237 J Warga Problems with delayed and shifted controls 251
- 14. ZARTSTEIN* Variational limits constrained by measure-valued multifunctions 1. Introduction The paper analyzes a variational problem depending on a parameter and examines the limit behavior when the parameter tends to a limit. The goal is to identify the variational limits of the problem, namely a variational problem which reveals both the limit value and the limit of optimal solutions. The paper demonstrates how measure-valued mul tifunctions arise as constraints in the variational limits, when the con straints in the original problem depend on highly oscillatory terms. The solution to the variational limit is a measure-valued function. It is shown how to use the solution for the variational limit to generate near optimal ordinary solutions to the original parameterized problem. The specific problem we examine is the following parameterized al location problem: (i.i) The optimization is over the allocation function #(•), where the variable x is in Rn and the value function u is real valued. The function p€ *Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics. The research was supported by a grant from the United States-Israel Binational Science Foundation (BSF). 1
- 15. takes values in a separable complete metric space T; the subscript e is a parameter which for definiteness is taken to be in (0, SQ] . Denote by val(e) the optimal value of (1.1), namely the supremum of the payoff integral in (1.1) subject to the given constraints. The dependence of the problem on the parameter e is the subject of the present paper. In particular, we wish to examine the convergence of val(e), and the convergence of optimal solutions of (1.1) as e —• 0. The parameter e affects the constraint set-valued map F, which takes closed subsets of Rn as values. (At the expense of complicated no tations we could consider a value function u which depends on the parameter, see Remark 6.1; see Remark 6.2 for other parameter depen dence forms.) The technical assumptions on the data are displayed in the next section. Without the presence of the parameter function pe(-) the optimiza tion problem is a classical allocation problem encountered, for instance, in operations research and economics. The underlying space, which for convenience is chosen here to be the interval [0,1], is interpreted as a space of, say, agents, and the real valued function u indicates the utility of allocating the amount x(t) to the agent t. There is the quantity à to allocate, and the assignment of the (infinitesimal) quantity x(t) to the agent i, results in the (infinitesimal) utility t¿(í, x(t)). Hence the goal is to maximize the total utility, subject to the constraint à on the resources, and subject to the constraints on the allocation represented by F. For an analysis and applications of the allocation problem with out the presence of the parameter e see e.g., Yaari [27], Aumann and Perles [8], Berliocchi and Lasry [15], Artstein [2], Aumann and Shapley [9], Arrow and Radner [1]. The e appearing in the model (1.1) signifies a possible sensitive dependence of the problem's data on some, say small, parameter. The sensitivity of the dependence is depicted by, possibly, highly oscillatory behavior of p£(>) for e small. We are interested in the determination of the limit behavior of the values of (1.1) and its optimal solutions as e tends to 0. To this end we produce a variational limit, namely an 2
- 16. allocation problem associated with the limit as e tends to 0. The term "variational limit" refers to the situation where the optimal value of the limit problem is indeed the limit of the optimal value of (1.1) as e tends to 0, and the solution of the limit problem is, in the appropriate sense, the limit of solutions of (1.1) as e tends to 0. Furthermore, we show how the solution of the limit problem can be employed to produce near optimal solutions of the problem (1.1) when e is small enough. Variational limits of optimization and control problems, with roles similar to the description in the preceding paragraph, were identified in a variety of problems. The T-convergence plays a similar role, see e.g., Dal Maso [18] and references therein. It was applied to control problems which include the form (1.1), see e.g., Buttazzo and Cavazzuti [17], Belloni, Buttazzo and Preddi [14], and Freddi [19, 20]. The related notion of epi-convergence is also designed to capture the limit behavior, see Attouch [6], Rockafellar and Wets [23]. For control problems, [4] produces variational limits based on the Young measure limits of the parameter function Pe(-). (Young measure solutions, also called relaxed solutions, to variational and control problems have been known for a long time; see e.g., Young [28], Warga [26], Valadier [25], Balder [11], Ball and Knowles [13]. It is the Young measure limit of the parameters that is the novelty in [4].) In this paper we employ the Young mea sure limit of the parameters for the parameterized problem (1.1). This results in a variational limit with a multimeasure constraint set. The purpose of this paper is to demonstrate how such a limit is generated and how to exploit its variational limit characteristics. The rest of the paper is organized as follows. In Section 2 we provide the technical assumptions on the data and recall some preliminary facts, primarily on Young measures. Section 3 applies the limit process to the parameter functions pe and (with the aid of a result in [5]) singles out the variational problem which arises when the limit is taken. This limit problem has measure-valued constraints and measure valued solutions. In Section 4 we establish, under appropriate conditions, two ingredients of the variational limit, namely the convergence of the values and the convergence of the solutions. We find that the convergence of solutions 3
- 17. holds is in the weak or the narrow topology. The third ingredient, namely the generation of near optimal solutions, is examined in Section 5. Some comments on the model, in particular on the need for some of the conditions, are listed in Section 6. In the closing section we indicate, via an illustrative example, how the limit problem can be analyzed and applied. 2. Preliminaries In this section we display the technical assumptions on the data ap pearing in the problem (1.1), including assumptions on the growth of the utility function u and the constraints map F. We also recall some facts concerning the narrow convergence of Young measures. The choice of the interval [0,1] as the space underlying the inte gration is made for convenience of notations; any complete separable metric space with an atomless underlying probability measure on it will do (see also Remark 6.3 for the possibility of using a measure space without a topology). Whenever encountered with a measurability structure on a metric space, the Borel field is considered. We restrict the discussion to de cisions in the non negative orthant i?+ of i?n , namely x is assumed greater than or equal to 0 coordinate-wise (see Remark 6.4). Likewise, the inequality in (1.1) is interpreted coordinate-wise. We write x > 0 if each coordinate of x is strictly bigger than 0. We write x for the euclidean norm of x. The Lebesgue measure on [0,1] is denoted by À. For each fixed e the mapping pe(-) maps [0,1] into a given separable and complete metric space Y (and we denote the elements in T by 7). It is assumed that for each e the function p£(-) is measurable. Assumption 2.1. (i) The function u (which is defined on [0,1] x R)j is assumed measurable with respect to the variable t and con tinuous with respect to x. (ii) The function u is o(x) as x -+ 00 integrably in t (namely, for each 6 > 0, there exists an integrable func tion rj(t) such that u(tjx) < 6x whenever x > rj(t)). (Hi) The 4
- 18. valúes ¡Q u(t,x(t))dt are uniformly bounded for families #(•) which are bounded by an integrable function. The measurability and continuity assumptions on u are standard as sumptions, yet for many applications it suffices to assume upper semi- continuity in x. This is not the case in the present paper, as demon strated in Example 6.5. The growth condition o{x) was introduced by Aumann and Pedes [8, Definition 1] to guarantee existence of solutions to the problem (1.1) with the constraint set F being equal to Rj¡_. It is possible to show (see Remark 6.6) that under Assumption 2.1 for every e > 0 the problem (1.1) has a solution, say #£(-), provided that the problem is feasible, namely that there exists a measurable selection #(•) of F(-,p£(-)) such that ¡Q x(t)dt < â. We are interested in the limit of this solution as e tends to 0. The growth condition plays a role in our analysis beyond guaranteeing the existence (see Example 6.7). In what follows we consider the space of closed subsets of Z£n , de noted by C(Rn ); we are actually employing C(Rr ¡_). We consider C(Rn ) with the Kuratowski convergence, namely C = Lim C¿ if every cluster point of a sequence qfc G C¿fc is in C, and each point c in C is the limit of a sequence c¿ in C¿. The Kuratowski convergence on closed subsets of Rn is metrizable, and the metric is separable and complete (see, e.g., [21]). The constraints set-valued mapping F(i, 7) is defined on [0,1] x T and takes closed subsets of i?+ as values; namely, its values are in C(Rn ). We assume that F is measurable in the variable t and con tinuous in the variable 7. For a fixed e the integral of the set-valued map F(-,pe(-)) forms the feasible set for the optimization (1.1). We postpone the discussion of the feasible set to Section 4, where we define also the analogous set for the limit problem. The following assumption is made in order to avoid some pathologies (see Example 6.8): Assumption 2.2. The family of functions ¿£(t) = min{x : x G F(t,pe(t))} 5
- 19. is uniformly integrable. Next we recall the notions of Young measure and narrow conver gence. For a general overview and applications see Balder [11], Val- adier [25]. Let S be a complete separable metric space. We denote by V(S) the space of probability measures on 5, endowed with the metric of weak convergence of measures; the space V(S) is then a complete separable metric space, see Billingsley [16]. A Young measure is a mea surable mapping, say v = */(•), from [0,1] into V(S). We denote the space of Young measures by y = 3^([0,1], S). The mapping v can be considered as a probability measure on [0,1] x S determined by (2.1) As such, elements v are subject to the metric of weak convergence on V([0,1] x S). We consider the space ^([0,1] x S) with this metric, which we also call the metric of narrow convergence. A useful criterion for the narrow convergence of i/¿(-) to ^o(') is a s follows: (2.2) for any bounded function r : [0,1] x S —• R which is measurable in t and continuous in s. (The customary criterion requires (2.2) for r bounded and continuous; it is easy then to see that (2.2) holds also for r bounded, continuous in s and only measurable in i, this since v(E x S) = X(E) for each E C [0,1], see e.g. [5, Lemma 2.4].) A measurable mapping h(-) from [0,1] into S can be viewed as a Young measure which assigns to each t the Dirac measure ^h(t) ? n a m e l y the probability measure supported at {h(t)}. The narrow convergence applied to point-valued maps may result in a Young measure limit. 3. The variational limit With the aid of the narrow limit of the parameter pe(-) in (1.1), we exhibit in this section a candidate for a variational limit for (1.1). We 6
- 20. indicate how the suggested limit arises naturally, however, after over coming a conceptual problem, which is resolved with the aid of a result established in [5]. In the next two sections we show that indeed a variational limit was obtained. Assumption 3.1. The functions p£(-) appearing in (1.1) converge narrowly to a Young measure, say to /i(-) = /x. See Remark 6.9 on the extent of the assumption. Next we display the variational limit, then define and explain the terms it refers to. (3.1) where M(t) is the collection of probability measures on RJ. selec- tionable with respect to the probability measure on S(R) given by F(t,/x(t)). In analogy with the problem (1.1) we denote by val(O) the optimal value of (3.1). A solution of (3.1) should be a Young measure £ = £(•) with values in V{Rn ). Thus, the form of (3.1) is as indicated in the title of the paper. We explain now the terms used in (3.1). For the probability measure fi on T, the probability measure F(t, ¡i) on C(R) is (for t fixed) the measure induced from /i by F(t, 7), namely F(t,n)(A) = n({1:F(t,7)€A}). Since F is assumed continuous in 7 it is clear that for t fixed F{t,n) is indeed in V(C(Rn )). It is also clear that since //(•) is a Young measure into the space T, then the mapping F(-, //(•)) is a Young measure into 7
- 21. C(Rn ). It is true (see Lemma 4.2) that the the narrow convergence of pe(-) to //(•) implies the narrow convergence of JP(-,pe(-)) to F(-, //(•))• Let 1/ be a probability measure on C(Rn ), and let p be a probability measure on Rn . The measure p is selectionable with respect to v if there are a probability measure space (iî, £, À), a measurable set-valued map H from £î into C(Rn ) and a measurable point-valued map h from fi into i?n , such that /¿(u;) 6 #(u;) for all a;, Dh = p and D if = i/, where D/¿ and DH are the probability distributions induced by h and £T, namely Dh(A) = ({u;:h(v)eA}), and likewise for H. Notation 3.2. For a probability measure v on C(i?n ), denote by S{y) the family of probability measures p on i?n which are selectionable with respect to v. (Thus, in (3.1) M(t) = S(F(t, n(t))).) Remark 3.3. The family S(u) for v in VC{RJ+), was characterized in [3], with a slight extension offered in [5] (See also Ross [24]). The family coincides with the collection of probability measures p satisfying p(K)<v{{C:CDK¿Q}) for every compact set K C Rn . In particular, S(v) is a convex set of probability measures, and it is not difficult to show that it is closed with respect to the weak convergence. The heuristics behind the construction of the limit problem (3.1) is that as e tends to 0, the optimal solutions of (1.1), say xe(-), may converge as well; but since the parameter functions p£(-) converge only narrowly, the most we can hope for is that xe(-) will converge narrowly. Hence the limit should be a Young measure on i?!f.. This clearly ex plains the expressions for the utility and for the integral constraint in the limit problem (3.1). A natural restriction on the domain of £(t) should actually have been £(t)eF(t,Ai(t)) (3.2) 8
- 22. with F(t, fi(t)) being the induced probability measure as explained ear lier. But, there is a syntactic problem with the form (3.2). Indeed, the relation "#(•) is a selection of F(-,pe('))" required in (1.1), loses it meaning when it comes to "£(•) is a selection of F(-, //(•))" written in (3.2). The difficulty is in the requirement that a probability distribu tion on points be a selection of a probability distribution on sets. The problem was addressed in [3],[5], where the selectionable probabilities as defined earlier were analyzed and employed. 4. The convergence This section verifies that the optimization problem (3.1) is a variational limit of (1.1) as e tends to 0, in the sense expressed in the following theorem. In the next section we show how a solution of (3.1) gives rise to near optimal solutions of (1.1) for e small. Recall that we work under Assumptions 2.1, 2.2 and 3.1. Theorem 4.1. Suppose that a fixed open neighborhood of the re source vector à is in the feasible set of (1.1) for e small enough. Then à is in the feasible set of (3.1), and the optimal values val(e) of (1.1) converge as e tends to 0, to the optimal value val(0) of (3.1). Further more, let #*(•) for e fixed be an optimal solution of (1.1). Then for any sequence S{ converging to 0, there exists a subsequence Sj such that #£.(•) converge narrowly to a solution £(•) of (3.1). A term which is self explanatory, yet may need a formal definition, is the feasible set. To this end recall the integral of a set-valued function, say H from [0,1] into C(i?+), given by (see Aumann [7] and Klein and Thompson [21]): (4.1) We call Jo F(t> Pe{t))dt the exact feasible set for the problem (1.1) with e fixed. The apparent reason is that the integral contains the resources 9
- 23. vectors that can be allocated. The feasible set is the set which contains all the vectors ÜQ such that there exists an a1 in the exact feasible set such that af < ÜQ. The assumptions in the theorem imply that the feasible set contains actually an a! independent of e such that a1 < à (see Example 6.10 for the need of this condition). The exact feasible set is convex ([7], [21]), but may not be closed. The feasible set is always closed, in view of the fact that the integral of a closed-valued set-valued map contains the extreme points of its closure (see, e.g., [2]). In the spirit of (4.1) we define the integral of a set-valued map G{t) into the space of probability measures on RJ+ as follows: (4.2) There is a slight abuse of formality in (4.2) as integrating sets of probability measures results with a subset of Rn . Since we do not need in this paper to integrate the measures in their own space, the use of (4.2) should not lead to a confusion. It is clear that (4.2) reduces to (4.1) when G(t) contains only Dirac measures, namely probability measures supported on a singleton. We call ¡Q M(t)dt the exact feasible set for (3.1). Due to the con vexity and closedness of M(t) it is easy to see that the exact feasible set is a convex closed subset of i?+. The feasible set of (3.1) consists, again, of the vectors CLQ such that there exists an af in the exact feasible set of (3.1) such that a' < aç). The verification of the theorem will follow several observations, some technical and some of interest for their own sake. Lemma 4.2. Since p6(-) converge narrowly (as Young measures into T) to //(•) it follows that F(-,pe(-)) converge (as Young measures intoC(Rn )) to F(-, //(•)). Proof. The result follows since by the continuity of F in the 7 variable, any bounded continuous function r : [0,1] x C(Rn ) — > i?, can 10
- 24. be interpreted as a continuous bounded function r : [0,1] x T — » i?, hence the criterion (2.2) for convergence can be verified. The following is a key step in our proof. It follows from a more gen eral result established in [5]. Recall that M(t) is the set of probability measures selectionable with respect to ¡i(t). Proposition 4.3. The ensemble S(F(-, pe(-))) of measurable selec tions converge in the Kuratowski sense with respect to narrow conver gence, to the ensemble <S(M(-)) of Young measures selectionable with respect to M(-); namely, whenever x€i(-) for S{ -> 0 are measurable se lections ofS(F(-,p£i(-))) which converge narrowly to a Young measure £(•), then £(•) is in S(M(-)), and for every £(•) in <S(M(-)) there are selections xe(-) of S(F(-, p€(-))) which converge narrowly as e tends to 0,to£(-). Proof. Given Lemma 4.2, the result is a particular case of [5, Theorem 6.1]. The narrow convergence by itself does not imply the quantitative results sought after in the theorem. To establish these we need some es timates which result from the assumptions imposed on the data. These estimates are verified in the following results. Notation. For each N > 0 and x G i?+ let N(x) be the vector whose i-th coordinate is equal to the i-th. coordinate X{ of x if X{ < N, and equal to N if X{ > N. In particular N(u(t, x)) = min(i¿(í, x),N). Lemma 4.4. A family xe(-) of selections in S(F(-,p£(-))) such that ¡Q,xe(t)dt < a for every e, forms a precompact family with respect to the narrow convergence, namely every sequence has a subsequence which converges narrowly. Proof. It is enough to verify tightness of the corresponding mea sures (see [16]). Tightness follows since the values of the functions are in R7 ^ and for any N > 0 the estimate (4.3) clearly holds. n
- 25. Proposition 4.5. Let x£(-) be selections of S(F(-,pe(-))) which converge narrowly as e tends to 0, say to £(•). Suppose also that for every e the estimate ¡Q xe(t)dt < ã holds. Then (4.4) and (4.5) as e tends to 0. Proof. The estimate (4.4) reflects the standard lower semiconti- nuity of the norm with respect to weak convergence. Here is a formal proof. For a given 6 let N be large enough so that (4.6) Such an N exists since the left hand side of (4.4) is finite. By the narrow convergence (see (2.2)) (4.7) the convergence being as e tends to 0. Since each term in the sequence in (4.7) is less than or equal to â, and since 6 is arbitrarily small, the estimate (4.4) follows from (4.6) and (4.7). To prove (4.5) let r?(-) be given by Assumption 2.1 for 6 > 0 small, and let N be such that (4.6) holds and (4.8) when 6(t) = 0 if N > t](t) and 6{t) = max{«(i,a;) : N < x < rj(t)} if N < r)(t). For any given 6 > 0 such an N exists by Assumption 2.1. 12
- 26. By narrow convergence the convergence in (4.5) holds if u is replaced by N(u), (see (2.2), and recall that u is assumed continuous in x). But in view of Assumption 2.1 and the construction of N (i.e.. (4.8)) changing u in (4.5) to N(u) results in a deviation (in both sides) of at most 6(1 + â). Since 6 is arbitrarily small the convergence in (4.5) is verified. This completes the proof. Lemma 4.6. Let h(-) and g(-) be integrable functions from [0,1] into Rn and let a be between 0 and 1. Then there exists a measurable subset, say A, o/[0,1], whose Lebesgue measure is equal to a, such that (4.9) Proof. Without the requirement that À(A) = a, the claim fol lows from the Liapunov Convexity Theorem. The additional condition À(A) = a can be deduced when the same argument is applied to the pair of functions (#(•)? 1) and (/¿(-),0). The following result can be viewed as a generalization of Balder [10, 12]. Proposition 4.7. Let £(•) be a selection of M(-) for which (4-4) holds. Then there exists a sequence xe(-) of selections of F(-,p£(-)) which converge narrowly to £(•) and such that (4.10) for every e, and furthermore. (4.11) Proof. By Proposition 4.3 there exist selections, say %(•) of F(-, Pe(-)) which converge narrowly to £(•). we shall modify them so to 13
- 27. get the desired sequence xe(-). We first aim at modifying ze{-) and get selections, say ye(-), which converge narrowly to f (•) and (4.12) (the last inequality is interpreted coordinate wise). To this end note that standard selection theorems (see e.g., [22]) together with Assump tion 2.2 imply the existence of selections of F(-, Pe(*)), say w£(-) which are uniformly integrable. For each N > 0 define ye N(Í) = z£(t) if |%(*)| < N &n d Ve N(t) — w e(t) otherwise. For every N the conver gence in (4.7) holds when z£ replaces x£. In particular for e fixed, if N is large enough we can deduce that (4.13) with 6(e) tending to 0 as £ tends to 0. If e is small enough, the narrow distance between z£(-) and £(•) is small, hence for e small enough and N large enough the Lebesgue measure of {t : ye jy(t) = w£(t)} is small. In particular, a relation N = N(e) can be determined such that ye N(e){m ) converges narrowly to £(•), and (in view of (4.13)) the claimed estimate (4.12) holds for ¡/e(-) = y6iN(e){-). Consider now a sequence, say g£(-) of selections of F(-,pe(-)) such that (4.14) where o! < a, (the existence of such a! is guaranteed by the conditions in Theorem 4.1). We use Lemma 4.6 to form a selection x£,a(') such that xe,a{t) = ye{t) for t in a set of Lebesgue measure greater or equal to (1 — a), and then when a is fixed and e is small enough the condition (4.15) 14
- 28. holds. However, when e is small, the parameter a can also be chosen small while maintaining the estimate (4.15). In particular, a relation a = a(e) can be determined such that xe a(e)(-) is the desired sequence. This completes the construction of selections satisfying (4.10). The requirement (4.11) follows from (4.5) in Proposition 4.5. Proof of Theorem 4.1. Let #£(•) be optimal solutions of (1.1). By Lemma 4.4 each sequence S{ has a subsequence Sj such that x^(-) converge narrowly, say to £(•). By Lemma 4.3 and (4.4) of Proposition 4.5 it follows that £(•) is feasible for the limit problem (3.1). By (4.5) of Proposition 4.5 (4.16) We show now that the right hand side of (4.16) is actually equal to val(0). To this end let £*(•) be an optimal solution of (3.1). By Propo sition 4.7 there are feasible selections x£(-) of (1.1) such that (4.17) from which we deduce that liminf val(e) > val(0). (4.18) The latter together with (4.16) imply lim val(e) = val(0). (4.19) This verifies the claim that the optimal values converge, and in view of (4.16) it follows that the narrow limit £(•) is indeed an optimal solution of (3.1). This completes the proof. 5. Near optimal solutions A desired feature of a variational limit is the possibility to use its solution to generate approximate, or near optimal, solutions for the 15
- 29. original problem. In this section we indicate how to carry out such a program with respect to (1.1) and (3.3). The suggested scheme is as follows. Proposition 5.1. Let £*(•) be an optimal solution of (3.1). Then there exist selections x£(-) o/F(-,pe(-)) which satisfy ¡Q x£(t)dt < à (namely they are feasible for (1-1)), and which converge narrowly to £*(•). For these selections the following holds. (5.1) Proof. Existence of the selections x£(-) was established in Propo sition 4.7. Prom property (4.5) of Proposition 4.5 it follows that Jo u(t, x£(t))dt converge to val(O), and since val(O) is the limit of val(e) as e tends to 0, the desired estimate (5.1) follows. Remark 5.2. The existence of near optimal solutions, as stated in the previous result, is implicit. We note that the construction of selections x£(-) which converge narrowly to a prescribed Young measure can be done using standard arguments in probability. The proof of Proposition 4.7 provides an explicit way to modify such selections and come up with the near optimal selections as in the previous proposition. 6. Comments Remark 6.1. In the problem (1.1) the payoff function u is indepen dent of the parameter e. It is possible to develop the theory with such dependence; some modifications are then needed. One apparent modi fication is to make the growth conditions on the value function, given in Assumption 2.1, uniform in the parameter e. The second, more in tricate, modification is to allow the limit payoff function to depend on a measure valued parameter (in addition to having the solution itself be a Young measure). Such considerations were developed in [4] for control problems without parametrized constraints. Remark 6.2. The parameter e in (1.1) enters through the param eterized function Pe(-)> namely the constraints are given in the form 16
- 30. F(í, Pe(í)). In particular, the narrow limit is examined on the functions Pe(-). This form was chosen for the convenience of notations only. We note here that such a form does not reflect any loss of generality with respect to the standard form of parameter dependence, namely using the form Fe(t) for the parametrized constraints. The conversion from the latter form to a representation by a mapping p£ can be done by letting pe(t) = F£(t). Such considerations are elaborated upon in [4], where the same equivalence is noted even for state dependent functions. Remark 6.3. The underlying space we use for the problem (1.1) is the unit interval, but as mentioned, any separable complete metric space will do. It should be noted that employing the setting worked out in Balder [11], would make it possible to consider the integration in (1.1) on an underlying space without a topological structure, namely on a general atomless measure space. Remark 6.4. We follow Aumann and Perles [8] in assuming that x is in the non-negative orthant of Rn ; it is clear, however, that this assumption may be relaxed. In fact, all the results hold when the values of set-valued function F are bounded from below by an integrable function into Rn . Some bound from below is, however, needed. Example 6.5. For a non parametrized problem of the form (1.1) it is enough, hence customary, to assume that u is merely upper semi- continuous in the x variable (it is then called a normal integrand); see Aumann and Perles [8], Berliocchi and Lasry [15]; see also Rockafellar [22] for a general discussion on normal integrands. In the case of the present paper the continuity of u in the x variable is needed. As a counter example consider the case where x is real, ã = 1, u(t,x) = 1 for x = 1 and i¿(í, x) = 0 otherwise, j(t) = e and F(i, e) is [0,1 — e]. The limit problem (which does not involve measures in this case) is defined by F(t) = [0,1], but the values do not converge. Another place for the need of u continuous in x is when the near optimal solutions are constructed (see Section 5). The argument there is based on the nar row convergence of the approximations to the limit solution. Without 17
- 31. the continuity of u in the x variable there is no guarantee that a good approximation of the payoff is obtained. Remark 6.6. Aumann and Perles [8] established the existence of an optimal solution of (1.1) (with e fixed) when the constraint set F is identical to BJ+. When F is a set-valued function with values being closed subsets of i?+, we have first to assume that there exists at least one selection of F(-) that meets the constraints. Then the arguments in [8] which ensure the existence apply also in the present case. Alternatively, the existence result follows as a particular case of [2, Theorem 1]. Example 6.7. The growth condition o(x) of Assumption 2.1(ii) is needed not only for the existence of optimal solutions. Consider the case where x is a scalar, à = 1, u(t,x) = x, j(t) = £, and F(t,e) = {OjU^-^oo]. For every e an optimal solution exists and the optimal payoff is 1. The optimal solutions even converge narrowly, yet in the limit, the payoff is 0. The boundedness condition on JQ u(t,x(t))dt stated in Assumption 2.1(iii) cannot be dropped, as without it val(s) may be infinite and the continuity at 0 may not hold. This assumption follows when the growth condition (ii) holds and u(t,x) is monotone in x. Example 6.8. We show that Assumption 2.2 cannot be dropped (note that in the standard framework, e.g., as in [8], the assumption is fulfilled in a trivial way as 0 is a feasible selection). Let x be scalar, à = 1, t¿(í, x) = min(x, 1), 7 = £, and let F(t, e) = {0,1} if e < t < 1 and F(t,e) = [e:_1 ,oo] when 0 < t < e. For e fixed, the only way to satisfy the constraints results in a payoff equal to e. In the limit problem a payoff 1 can be obtained. Remark 6.9. The prime advantage of Young measures is that compactness, namely existence of converging subsequences, holds under mild conditions. This is the rational behind employing Young measures as a solution concept. See Young [28], Warga [26], Valadier [25] and Balder [11]. Compactness is equivalent to: For each 6 there exists a compact set K in V such that ({t : p£(t) £ K}) < 6. 18
- 32. Example 6.10. We show that the existence of the vector a1 < à in the feasible set of (1.1) for e small enough, cannot be dropped from the assumptions of Theorem 4.1. Consider x = (#i,£2) in R2 , let à = (1,1), u(t, x) = min(:ri, x2), J = e, and F(t, e) = {(2,0), (0,2), (1+ £, 1 + e)}. Then à is in the feasible set of (1.1) for every e > 0, but the payoff is 0. The optimal payoff in limit problem is clearly 1. 7. An illustrative example This section exhibits an example that besides serving as an illustration to the abstract result, indicates how the variational limit can be solved. To this end we start with a characterization of solutions to the limit variational problem. For simplicity, the conditions are stated under assumptions on the data stronger than previously in the text. Proposition 7.1. Suppose that the utility function u(t,x) is non- decreasing in the variable x. Suppose also that the vector à is in the exact feasible set, namely a selection v(-) of M(-) exists, such that (7.1) Then a necessary and sufficient condition for a measurable selection */*(•) of M(-) for which (7.1) is satisfied, to be an optimal solution of (3.1) is the existence of a vector q > 0, such that (7.2) for almost every t G [0,1]. Proof. The condition (7.2) is the Kuhn-Tucker condition for the problem (3.1). Compare with Aumann and Perles [8, Theorem 5.1]. 19
- 33. Indeed, the convexity of the set M(t) implies that the set of points in given by (7.3) is a convex set. A point on the boundary of this set, in particular the point generated by an optimal solution, is characterized by a supporting hyperplane to the set generated by a vector of the form (1, —q) with q > 0. Due to the integral form in t the same q serves as a supporting vector for almost all t. Example 7.2. Let a; be a scalar. Let u(t,x) = Un(x + 1) (in being the natural logarithm). Let 7 take two values, say +1 and —1. Let Fit,')) be the set of non negative even numbers if 7 = 1, and let F(t, 7) be the set of positive odd numbers together with 0 if 7 = — 1. Let p£(t) — sign cos(e t). (Namely, in the allocation process, unless 0 is assigned, only even or odd values can be assigned, according to whether cos(e~1 t) is positive or negative). Finally, let à > 0 be fixed. It is clear that p£(-) converges narrowly to the Young measure which does not depend on time, and assigns equal probabilities to +1 and —1. Accordingly, the set M(t) of selectionable distributions of the problem (3.1) associated with the present example, is independent of i, and consists of all the probability measures on the integers which assign probability less than or equal to ^ to both the set of positive even numbers and the set of odd numbers. For a given q > 0 we can determine the measure /i(g) which gives rise to the maximum according to the criterion (7.2). Indeed, for q fixed, with the exception of at most a denumerable number of t the optimal measure ¿¿(i, q) is as follows. (7.4) 20
- 34. here [t/q] is the largest integer which is less than or equal to t/q. The multiplier q associated with the optimal solution is then determined by the equality (7.5) The preceding equality can be made more concrete as follows. For a given q > 0, let n = [t/q], namely n is the largest integer such that nq < 1 and (n + l)q > 1. Then (7.5) reads as follows. (7.6) For a given à the solution of (7.5) (or (7.6)) is unique. Indeed, the left hand side is increasing as q decreases. For a given à the multiplier q can be determined through the equa tion (7.6), and the optimal solution for the variational limit is then the Young measure given in (7.4). Near optimal solutions for small e are then determined by a narrow limit of this Young measure. A direct inspection shows that a near optimal solution is given by (7.6), x£(t) is the even or odd number among max([i/g] — 1,0) and max([t/g] — 2], 0), according to whether sign cos(e-1 i) is +1 or — 1. References [1] K. J. Arrow and R. Radner, Allocation of resources in large teams. Econometric^ 47 (1979), 361-385. [2] Z. Artstein, On a variational problem. J. Math. Anal Appl. 45 (1974), 404-415. [3] Z. Artstein, Distributions of random sets and random selections. Is- rael J. Math. 46 (1983), 313-324. [4] Z. Artstein, Chattering variational limits of control systems. Forum Mathematicum 5 (1993), 369-403. [5] Z. Artstein, Relaxed multifunctions and Young multimeasures. Set- Valued Anal, to appear. 21
- 35. [6] H. Attouch, Variational Convergence of Functions and Operators. Pitman, Boston, 1984. [7] R. J. Aumann, Integrals of set valued functions. J. Math. Anal. Appl. 12 (1965), 1-12. [8] R. J. Aumann and M. A. Perles, A variational problem arising in economics. J. Math. Anal. Appl. 11 (1965), 488-503. [9] R. J. Aumann and L. S. Shapley, Values of Non Atomic Games, Princeton University Press, Princeton, 1974. [10] E. J. Balder, A general denseness result in relaxed control the ory. Bull. Australian Math. Soc. 30 (1984), 463-465. [11] E. J. Balder, Lectures on Young Measures. Cahiers de Mathématiques de la Décision 9517, CEREMADE, Université Paris-Dauphine, Paris, 1995. [12] E. J. Balder, Consequence on denseness of Dirac Young measures. J. Math. Anal. Appl. 207 (1997), 536-540. [13] J. M. Ball and G. Knowles, Young measures and minimization prob lems in mechanics. In: Elasticity, Mathematical Methods and Appli cations, G. Eason and R. W. Ogden eds., Ellis Horwood, Chichester 1990, pp 1-20. [14] M. Belloni, G. Buttazzo, and L. Preddi, Completion by gamma- convergence for optimal control problems. Ann. Fac. Sci. Toulouse Math. 2 (1993), 149-162. [15] H. Berliocchi and J.-M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973), 129-184. [16] P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968. [17] G. Buttazzo and E. Cavazzuti, Limit problems in optimal control theory. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), suppl., 151-160. [18] G. Dal Maso, Introduction to T-Convergence. Birkhauser, Boston, 1993. 22
- 36. [19] L. Freddi, Variational limits of optimal control problems. Research Report, University of Udine, 1997. [20] L. Preddi, T-convergence and chattering limits in optimal control theory. Research Report, University of Udine, 1998. [21] E. Klein and A. C. Thompson, Theory of Correspondences. Wiley- Interscience, New York, 1984. [22] R. T. Rockafellar, Integral functional, normal integrands and mea surable selections. In: Nonlinear Operators and the Calculus of Variations, L. Waelbroeck, éd., Lecture Notes in Mathematics 543, Springer-Verlag, Berlin, 1976, 157-208. [23] R. T. Rockafellar and R. J-B Wets, Variational Analysis, an in troduction. In: Multifunctions and Integrands, Stochastic Analysis and Optimization, G. Salinetti éd., Lecture Notes in Mathematics 1091, Springer-Verlag, Berlin 1984, 1-54. [24] D. Ross, Selectionable distributions for a random set. Math. Proc. Cambridge Phil. Soc. 108 (1990), 405-408. [25] M. Valadier, A course on Young measures. Rend. Istit. Mat. Univ. Trieste 26 (1994) supp., 349-394. [26] J. Warga, Optimal Control of Differential and Functional Equa tions. Academic Press, New York, 1975. [27] M. E. Yaari, On the existence of optimal plan in continuous-time allocation process. Econometrica 32 (1964), 576-590. [28] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, New York, 1969. Zvi Artstein Department of Theoretical Mathematics The Weizmann Institute of Science Rehovot, ISRAEL e-address: zvika@wisdom.weizmann.ac.il 23
- 37. 1 E J BALDER New fundamentals of Young measure convergence Introduction This paper presents a new, penetrating approach to Young measure convergence in an abstract, measure theoretical setting. It was started in [12, 13, 14] and given its definitive shape in [18, 22]. This approach is based on K-convergence, a device by which narrow convergence on V(Rd ) can be systematically transferred to Young measure convergence. Here ^(R^) stands for the set of all probability measures on Rd (in the sequel, a much more general topological space S is used instead of Rd ). Recall that in this context Young measures are measurable functions from an underlying finite measure space (iî,,4,fi) into V(Rd ). Recall also from [12], [13] (see also [24]) that 7-convergence takes the following form when applied to Young measures (see Definition 3.1): A sequence (6k) of Young measures K-converges to a Young measure ¿o [notation: 6k —• ¿o] if f°r every subsequence (6^) of (6k) the following pointwise Cesaro-type convergence takes places at //-almost every point UJ in H. Here "=>" means classical narrow convergence on T^R^) (see Definition 2.1). As is shown much more completely in Proposition 3.6 and Theorem 4.8, the following fundamental relationship holds between Young measure convergence, denoted by " = > " , and K-convergence as just defined [18, Corollary 3.16]: Theorem 1.1 Let (6n) be a sequence of Young measures. The following are equivalent: (a) 6n = > 60. (b) Every subsequence (6n>) of (6n) contains a further subsequence (6n») such that 6n" — • ^o- 24
- 38. Both the nature of this equivalence result and the way in which we shall employ it are rather reminiscent of the well-known characterization of convergence in measure in terms of convergence almost everywhere. But while the latter result is simple, the for mer one is deep, as will become clear in the sequel. Nevertheless, thanks to this result several fundamental results on (sequential) Young measure convergence become simple to derive and can be stretched to what are arguably their most general versions in an abstract setting. These include: (1) the Prohorov-type criterion for relative sequential narrow compactness (Theorem 4.10), (2) the support theorem (Theorem 4.12, (3) the lower closure theorem (Theorem 4.13), (4) the denseness theorem for Dirac Young measures. The power of the apparatus thus developed is demonstrated by a selec tion of advanced applications in section 5, some of which are new as well (see also [18, 22] for references to applications in economics, such as [19, 21]). To the interested reader we also recommend [44, 48, 49, 56, 57, 59] for further background material and orientations towards various applications in applied analysis and optimal control. 2 Narrow convergence of probability measures This section recapitulates some results on narrow convergence of probability measures on a metric space; cf. [2, 27, 28, 35, 46]. Let 5 be a completely regular Suslin space, whose topology is denoted by r. On such a space there exists a metric p whose topology TP is not stronger than r, with the property that the Borel cr-algebras B(S,TP) and B(S,T) coincide. To see this, recall that in a completely regular space the points are separated by the collection Cb(S^r) of all bounded continuous functions on 5. Since S is also Suslin, it follows by [32, III.32] that there exists a countable subset (c¿) of Cb(S, r), with sup^s |ct(^)| = 1 for each ¿, that still separates the points of 5. A metric as desired is then given by p(x,y) := £ S i 2~l ci(x) — C{(y). This is because rp C r is obvious and by another well-known property of Suslin spaces, the Borel cr-algebras B(S,p) and B(S,T) coincide [51, Corollary 2, p. 101]. Of course, if 5 is a metrizable Suslin space to begin with, then for p one can simply take any metric on S that is compatible with r. As a consequence of the above, we shall write from now on for respectively the Borel cr-algebra and the set of all probability measures on (5, B(S)). Definition 2.1 (narrow convergence in V(S)) A sequence (vn) in V(S) converges Tp-narrowly to u0 £ V(S) (notation: vn => v>0) if limn fs c dvn = fs c di/0 for every c in Cb{S,rp). Here Cb(S,Tp) stands for the set of all bounded rp-continuous functions from S into R. Although Tp-narrow convergence is more fundamental for our purposes, we shall often be able to use the stronger form of narrow convergence that arises when Cb(S, rp) in the 25
- 39. above definition is replaced by the larger set Cb(S,r). This will be denoted by " = S > ". Definition 2.1 obviously extends to a definition of the r- and rp-narrow topologies on V(S); we indicate these by TT and Tp). By [51, Appendix, Theorem 7] V(S) is a Suslin space for TT (it is also Suslin and even metrizable for Tp - cf. [35, 111.60]). Hence, completely analogous to what was observed above for 5, the Borel cr-algebras coincide by [51, Corollary 2, p. 101]: B(V(S)) := B(P(S),TT) = B(V(S),TP). A vehicle by which we frequently manage to go from ^-convergence to the more general r-convergence is r-tightness: Definition 2.2 (tightness in V(S)) A sequence (i/n) in V(S) is said to be r-tight if there exists a sequentially r-inf-compact function h : S —• [0, +oo] (i.e., all lower level sets {x € S : h(x) < ft}, ft € R, are sequentially r-compact) such that supn fs h dvn < -foo. Observe that a fortiori h must be rp-inf-compact on S (causing h to be Borel measur able); note here that rp is metrizable, so that the distinction between sequential and ordinary rp-inf-compactness vanishes. Remark 2.3 The above definition can be shown to be equivalent to the following one [5, Example 2.5]: (Sn) is r-tight if and only if for every e > 0 there exists a sequentially r-compact set Ke such that supn vn(SKe) < e. Theorem 2.4 (portmanteau theorem for =>) (i) Let(i/n) and i/0 be inV(S). The following are equivalent: (a) vn £> i/0. (6) limn Js c dvn — ¡s c di/0 for every c eCu(S,p). (c) lim infn fs q dvn > fs q dv0 for every rp-lower semicontinuous function q : S — » (—oo,-foo] which is bounded from below. (ii) Moreover, if (vn) is r-tight, then the above are also equivalent to (d) i/n =£• I/Q. (e) lim infn fs q di/n > fs q dvQ for every sequentially r-lower semicontinuous func tion q : S —> (—oo,+oo] which is bounded from below. Here Cu(S,p) stands for the set of all uniformly /o-continuous and bounded functions from S into R. The name "portmanteau theorem" comes from [28]. Proof. Part (z), which is stated in a metrizable context, is classical; cf. [2, 4.5.1], [27, Proposition 7.21] and [28, Theorem 2.1]. Next, we prove part (ii): (d) =* (a) holds a fortiori, (e) => (d) is evident since for any c G Cb(S,r) both c and —c meet the conditions imposed on q in part (e). (d) => (e): Let h be as in Definition 2.2 For any q as stated in (e) and for any e > 0 the function qt := q + eh is sequentially 26
- 40. r-inf-compact, whence rp-inf-compact. Hence, qt is rp-lower semicontinuous on S and bounded from below.1 So (c) and an easy argument with e —• 0 give (e). QED It turns out that tightness is a criterion for relative compactness in the narrow topology. Just as in Definition 2.2 we only state the sequential version. Theorem 2.5 (Prohorov's theorem for =>) Let (un) in V(S) be r-tight. Then there exist a subsequence (vn>) of (i/n) and v* G V(S) such that vni = S > v+. Proof. By T D TP we can apply Prohorov's classical theorem [28, Theorem 6.1]. Hence, there exist a subsequence (vni) of (i/n) and v* G V(S) such that vn> => i/*. Hereupon, we can invoke Theorem 2.4. QED Let N := NU{oo} be the usual Alexandrov-compactification of the natural num bers. This is a metrizable space, so let p be a fixed metric on N and let S := S x: N. We can equip S with the product metric p or with the product topology f := r x r¿. For n € N, let cn € V(N) be Dirac measure concentrated at the point n. The proof of the next result is rather obvious by Theorem 2.4(6) and a triangle inequality argument. Corollary 2.6 Let (i/n) and v>0 be in V(S). / / ^ E Î L i ^n =^ u0 in V{S), then In particular, if vn => v0 in V(S), then Recall that the support r-supp v of v G V(S) defined to be the complement of the union of all open i/-null sets; hence, i/(r-supp v) = 1 (note that every r-open subset of 5 has the countable subcover property by [35, 111.67]). Theorem 2.7 (support theorem for =>) (i) Let ^E^=i v n =^ ^o for (un) and u0 in V(S) (in particular, this holds when vn •=$> u0). Then (ii) Moreover, if {vn) r-tight, then i/0(T-seq-cl r-LsnT-supp vn) = 1 and r-supp u0 C r-cl T-Lsnr-supp vni x This shows q : S — • R to be ^(5)-measurable, with q := q on {h < H-oo} and q := +oo on {h = +00}. Hence, the integrals in (e) are well-defined. 27
- 41. Here "r-seq-cl" stands for sequential closure with respect to the topology r and "r-Lsn" refers to the usual Kuratowski sequential r-limes superior of a sequence of subsets. This set is r-closed if r = rp (metrizable case). Proof, (i) By Corollary 2.6 it follows that TTN := jjYln=i{u n x tn) 4» u0 x e^ in V(S), where S := S x N. Setting Sn := rp-supp vn and Soo := rp-Lsnrp-supp vn, we define q0 : S -» {0,-foo} as follows: If x G Sk then q0(x,k) := 0. If x £ Sk for all k, 1 < k < oo, then ^o(^? &) := +oo. We claim that q0 is r^-lower semicontinuous in every point (x,k) of S x N. For let /5((arJ ',fcJ '),(a;, A:)) — » 0. We must show that a := liminfn qo(xj , P ) > q'0{x, k). If k < oo, then eventually F = A;, so a > qo(x, k) follows since S*; is rp-closed. If k = oo, we can have two cases: if eventually kJ ' = oo, then a > (jo(x, oo) follows by rp-closedness of S^. On the other hand, if F < oo infinitely often, then the same inequality follows directly from the definition of Soo- This shows that q0 is indeed Tp-lower semicontinuous. Now fg qod(vn x en) = Js <jo(x, n)i/n(dx) = 0 for every n. Hence, fsqodirjv = 0 for every N. Thus, Theorem 2.4 gives fsq0(x, oo)u0(dx) = 0, and the desired rp-support property for VQ follows. (iï) Since N is compact, f-tightness of (vn x en) in V(S) is evident. Hence, Theo rem 2.4 gives jj ^nzzii^n x en) 4> v0 x e^ in V(S). We now essentially proceed as in the proof of (¿), but a little more carefully: the additional sequential closure operation in the definition of S^) is needed because r-Lsnr-supp vn need not be sequentially r-closed on its own accord. QED Theorem 2.8 Lei vn =$> v0 in V(S). Then (vn) is rp-tight. Proof. 5 is Suslin, so any probability measure in V(S) is a Radon measure for both r and rp [35, III.69]. Hence, the result follows from [28, Theorem 8, Appendix III]. QED The above sufficient condition for rp-tightness of a sequence will play a role further on. It seems to have no analogue for r-tightness when r is nonmetrizable. The following result, also to be used later, is [27, Proposition 7.19]: Proposition 2.9 (countable determination of =>) There exists a countable set Co C { c G Cu(5, p) : sup5 |c| = 1} such that for every (i/n) and UQ in V(S) one has vn => i/0 if and only if lirrin fsc dvn = / 5 c dv0 for every c G Co. In particular, Co separates the points ofV(S). K-convergence of Young measures A Young measure is a function 8 : il — > V(S) that is measurable with respect to A and B(V(S)) The set of all such Young measures is denoted by K{tt; S). By B(S) = B(S)TP) of the previous section it is not hard to see that Young measures are precisely the transition probabilities from (ÎÎ,-4) i n t o (S,B(S)) [45, III.2], i.e., 6 : il -> V(S) belongs to K(Q; S) if and only if u> i-» S(u;)(B) is .4-measurable for every B € B(S). 3 28
- 42. For some elementary measure-theoretical properties of Young measures the reader is referred to [45, III.2] or [2, 2.6]. In particular, the product measure induced on {Six S, Ax B(S)) by p and any S G ft(ft; S) (cf. [45, III.2]) is denoted by fi ® 6. Let £°(iî;S) be the set of all measurable functions from (il, A) into (5, #(5)). A Young measure S G %(il S) is said to be Dirac if it is a degenerate transition probability [45, III.2], i.e., if there exists a function / € £°(il; S) such that for every UJ in Ü Conversely, S is also called the Young measure relaxation of / . In this special case 6 is denoted by S = ej. The set of all Dirac Young measures in 7£(í); S) is denoted by ^Dirac(^'i S). The fundamental idea behind Young measure theory is that, in some sense, 1Z(Ü; S) forms a completion of £°(i); 5), when the latter is identified with 1Zoirac(^'i S). Let us agree to the following terminology: an integrand on Í) x S is a function ^ : i î x S - > (—oo,+oo] such that for every w E Î Î the function g(u>, •) on S is B(S)- measurable. A function g : il x S — > (—oo, +oo] is said to be a (sequentially) r-lower semicontinuous [r-continuous] [[r-inf-compact]] integrand on Í) x S if for every wGÎÎ the function g(w, •) on S is (sequentially) T-lower semicontinuous [r-continuous] [[r- inf-compact]] respectively. Let g be an integrand on fl x S. The following expression is meaningful for any 6 G 7£(íl; S): provided that the two integral signs are interpreted as follows: (1) for every fixed u the integral over the set S of the function g(w, •), which is ¿?(5)-measurable by definition of the term integrand, is a quasi-integral in the sense of [45, p. 41], (2) the integral over Í7 is interpreted as an outer integral (note that outer integration comes down to quasi-integration when measurable functions are involved - cf. [9, Appendix A] or [22, Appendix B]). The resulting functional Ig : fc(il; S) —> [—oo, +oo] is called the Young measure integral functional associated to g. Another integral functional associated to g, this time on the set £°(i); S) of all measurable functions from Í1 into 5, is given by the formula The following notion of convergence was introduced and studied in a more abstract context in [12, 13]. Definition 3.1 (K -convergence in 7£(S1; S*)) A sequence (£n) in 7£(iî; S) K -conver ges with respect to the topology r to ¿>0 G 7£(iî; S) (notation: Sn —í- 60) if for every subsequence (Sn/) of (6n) 29
- 43. Note that in the expression above the exceptional null set is allowed to vary with the subsequence (6ni). We remark that À'-convergence is nontopological. If in the above definition r is replaced by rp and " = S > " by " => ", we obtain the weaker notion of K-convergence with respect to rp. This is denoted by " —^ ". We shall occasionally use u —> " in situations where we need not distinguish between the two at all. Example 3.2 Let (íl,.4,¿0 be ([0,1],£([0,1]), Ai) (i.e., the Lebesgue unit interval). Let (fn) be the sequence of Rademacher functions, defined by fn(u>) := sign sin(2n 7ru;) (here S := R, of course). Then tjn —• 60l where S0 G 7£([0,1];R) is the constant function S0((jj) = ~ti + |e_i. This can be proven by the (scalar) strong law of large numbers, analogous to the proof of Theorem 3.8. Definition 3.3 (tightness in K(il; S)) A sequence (Sn) in 7£(i); S) is r-tight if there exists a nonnegative, sequentially r-inf-compact integrand h on il x S such that supn Ih(Sn) < +oo. This definition comes from [26]; clearly this extends Definition 2.2. Recall from the previously given definition of integrands that a sequentially r-inf-compact integrand h is simply a function on Q x S with the following property: for every w G Q the function h(u),') is sequentially r-inf-compact. Remark 3.4 Similar to Remark 2.3, Definition 3.3 can easily be shown to be equiva lent to the following one [39]: (8n) is r -tight if and only if for every e > 0 there exists a multifunction I : fî — » 2s , with TC(LU) sequentially r-compact for every u; G ÎÎ, such that Example 3.5 (a) Let E be a separable reflexive Banach space with norm || • ||. Let E' be the dual space of E. Suppose that (/n) C ^(fyE) is £a -bounded: supn fu fn(uj)fi(du;) < +oo. Then (e/n) is a(E, Ef )-tight in 7£(Í7;5): simply set h(uû,x) := ||z|| in Definition 3.3. (b) Let E be a separable Banach space with norm || • ||. Suppose that (/n) C Cl (Çl',E) is /^-bounded and that there exists a multifunction R : Í) — > 2s such that for a.e. LJ both {/n(^) : n € N} C R(w) and R(u) is o~(E, ^^-ball-compact [i.e., the intersection of R(w) with every closed ball in E is cr(E, Ü^-compact]. Then (c/n) is o~(E,E')-tigh.t: now we set hn(uj,x) := x if x G -#(<*;), and /i^(o;,x) := -foo otherwise. Then for every u> € Q and ¡3 € R+ the set of all x G E such that hfi(uj, x) < P is the intersection of R{OJ) and the closed ball with radius ¡3 around 0. By the Eberlein-Smulian theorem it is sequentially cr(E, .¿^-compact as well. Part (6) in the above example generalizes part (a): simply observe that in part (a) E itself is a(E, E')-ball-compact by reflexivity, so there we can take R = E. A very important property of A'-convergence of Young measures is as follows [13, 12, 18]: 30
- 44. Proposition 3.6 (Fatou-Vitali for —> ) (i) Let Sn —^ S0 in 1Z(Ü;S). Then liminfn Ig(Sn) > Ig(So) for every rp-lower semicontinuous integrand g on H x S such that (3.1) (ii) Moreover, if(Sn) is r-tight, then also liminfn Ig(Sn) > Ig{60) for every sequentially T-lower semicontinuous integrand g on il x S such that (3.1) holds. Here, as usual, g~ := max(—#,0) and {g < —a}^ denotes {x G 5 : g(u,x) < —a}. Note that footnote 1 applies to each g(u;, •) in part (ii). Remark 3.7 If Sn = c/n for all n G N, then (3.1) runs as follows: Since g(íú,fn(uj)) < —a if and only if g~ (LÜ, fn(u>)) > a, (3.1) comes down to uniform (outer) integrability of the sequence (g~(-, fn('))) in the case of a Dirac sequence, in agreement with standard formulations; cf. [37, 5]. Proof of Proposition 3.6. The proof of (i) will be given in two steps. Step 1: g > 0. Set fi := liminfn Ig(Sn); then there is a subsequence (Sn>) such that P = Ymn. Ig(8n,). Define XI>N(U>) := }jT,n>=i Is 9(u,x)6n,(u;)(dx) and ^o(^) := Jsg((jü,x)ó0(u>)(dx). Then lim infjv I/>N > i¡)0 a.e. by Theorem 2.4(c), because by Defi nition 3.1 jfYln'=i àn'{u) 4> S0(LÜ) in V(S) for a.e. w. Thus, Fatou's lemma can be applied (it remains valid for outer integration in the direction that suits us; cf. [22, Appendix B]). This gives /? > liminf;v^oo JQ ^vcfy/ > JQ ipodfi = Ig(S0) by subadditivity of outer integration. Step 2: general case. We essentially follow Ioffe [37] by pointing out that J g(u,x)6n(u))(dx) + J l{g<-ct}(Lü,x)g-(uj,x)ón(u;)(dx) > J gQ(<jj,x)6n(uj)(dx), by ^r -f- l{g<_Qyg~ > ga := max(#, —a). One more (outer) integration gives, in the notation of (3.1), Ig(Sn) + s(a) > Iga(Sn), where we use subadditivity of outer integra tion. Now step 1 trivially extends to any g that is bounded from below, such as ga. This gives liminf Ig(6n) + s(a) > liminf Iga(6n) > I9a(S0) > Ig(60), where we use ga > g. The proof of (i) is finished by letting a go to infinity. (¿¿) Let h be as in Definition 3.3 and denote s := supn Ih(6n). We augment g, similar to the proof of Theorem 2A(ii): For e > 0 define gt := g--eh. Then gt > g and #c (u;, •) is Tp-lower semicontinuous on S for every UJ G 0 (see the proof of Theorem 2.4(H)). Thus, part (i) gives liminfn Ig(Sn) + es > liminfn Igc(6n) > Ig<(60) > Ig(60) for any e > 0. Letting e go to zero gives the desired inequality. QED The following important Prohorov-type relative compactness criterion for /^-conver gence is [13, Theorem 5.1]. It was obtained as a specialization to Young measures of an abstract version of KomlóV theorem [41]; see also [14]. 31
- 45. Theorem 3.8 (Prohorov's theorem for - ^ ) Let (Sn) in ft(ft; S) be r-tight A' T Then there exist a subsequence (6n>) of (Sn) and Sm G 1Z(Çl; S) such that 6n> —^ 6*. To prove Theorem 3.8 we use the following theorem, due to Komlós [41]. Theorem 3.9 (Komlós) Let (0n) be a sequence in Cx (ft; R) such that supn fQ (/>nd/ji < +oo. Then there exist a subsequence ((¡>nf ) of (<f>n) and a function </>+ G £1 (Í7;R) such that for every further subsequence (<j)n») of (<j>n') Lemma 3.10 Let (un) in V(S) be T-tight and let Co be a subset of {c G Cb(S,r) : sup5 c = 1} that separates the points ofV(S). If then there exists v* G V{S) such that vn = S > i/*. This lemma is a direct result of Theorem 2.5 and the point separating property of C0; cf. Proposition 2.9. Proof of Theorem 3.8. Let Co = {c, : i G N} be as in Lemma 3.10. Define <f>i,n(u) := fs Ci(x)6n(u>)(dx); then supn fQ |0¿,n|c?// < +°o for every i G N. Let h be as in Definition 3.3. By definition of outer integration, there exists for each n G N a function (j>o,n G £l (Q;R) such that (j>o,n{w) > fsh(uj,x)8n{uj)(dx) for a.e. UJ G O and In 0o,n dfx = Ih(8n). Applying Theorem 3.9 in a diagonal extraction procedure, we obtain a subsequence (Snt) of (8n) and functions <t>^m G £1 (Í);R), i G N U {0}, such that liiriiv ^ Z¡^/=i <^î,n" = <t>i,* a.e. for every further subsequence (6n») and for all i G N U {0}. Explicitly, we have every such (Snn) for a.e. u in Q (3.2) (3.3) Let us first consider (Sn>) itself as the subsequence in question. Fix to outside the exceptional null set M, associated with this particular choice of a subsequence in (3.2)-(3.3). Then (3.2) implies that the sequence (vN) in V(S), defined by vN := jjY^n'=iàn'(u)i is r-tight in the classical sense of Definition 2.2. Also, (3.3) implies that im.N ¡s Cidvjv exists for every i. By Lemma 3.10 there exists v^,* in V(S) such that z//v 4> UUJ^. Define ¿>*(u>) := i^u;,* for u G ÜM. Also, on M we define 6+ to be equal to an arbitrary fixed element of V(S). Then it is elementary to show that Sm 32
- 46. 4 belongs to 7£(iï;5). The argument following (3.3) can be repeated with a change of the null set M (for which Definition 3.1 leaves room) if one starts out with an arbitrary subsequence (6nn) of (6nt). QED The next example extends Example 3.2 and demonstrates the power of Theo rem 3.8, which brings K-convergence (for subsequences!) to settings where Kol- mogorov's law of large numbers, used in the special Example 3.2, stands no chance at all. Example 3.11 Let (iî,,4, //) be ([0,1],£([0, l]),Ai) (i.e., the Lebesgue unit interval). Let /i € >C1 ([0,1]; R) be arbitrary; it can be extended periodically from [0,1] to all of R. We define /n+i(u;) := /i(2n u;). Clearly, the sequence (e/n) is tight in the sense of Definition 3.3 (see Example 3.5(a)). By Theorem 3.8 there exist a subsequence (/n/) of (/n ) and some 6* £ 7£([0,1]; R) such that e/n, —• £*. The precise nature of 6m can now be determined by means of Proposition 3.6, but we shall defer this to Example 4.4 later on. The following are direct consequences of Corollary 2.6 and Theorem 2.7 for K- convergence of Young measures (by their application pointwise): Corollary 3.12 Let (Sn) and S0 be in 7£(fi;S). The following are equivalent: (a) 6n —^ ¿o in H(T; S) (b) 6n x en ^ So x Coo in H(T;S). Theorem 3.13 (support theorem for —•» ) (i) Let Sn —^ S0 in 1Z(Q;S). Then Tp-supp 60(u>) C Tp-LsnTp-supp Sn(uj) for a.e. UJ inVt. (ii) Moreover, if Sn —U 60, then also S0(u;)(r-seq-c r-Lsnr-supp 6n(w)) = 1, r-supp ¿o(k>) C r-cl r-Lsnr-supp Sn(uj) for a.e. u in 0. Narrow convergence of Young measures In this section our program to transfer narrow convergence results for probability mea sures (section 2) to Young measures comes is completed. We use the same fundamental hypotheses as in the previous section: (fi, .4, fi) is a finite measure space and (5, r) is a completely regular Suslin space, on which we also consider the weak metric topol ogy rp C T. We start out by giving the definition of narrow convergence for Young measures [3, 4, 10] (see also [38]). 33
- 47. Definition 4.1 (narrow convergence in 7l(T] S)) A sequence (Sn) in 1Z(Ü; S) con verges r-narrowly to S0 in 7£(fi; S) (this is denoted by Sn =^> S0) if for every A G A and c in C&(5, r) The weaker notion of rp-narrow convergence is defined by replacing r by rp this is denoted by " = > ". We shall occasionally use " = > " in situations where we need not distinguish between the two at all. We shall see that on r-tight sets of Young measures these two modes actually coincide (note the complete analogy to section 2). For further benefit, note carefully the distinct notation used for narrow convergence for probability measures (indicated by short arrows) and Young measure convergence (indicated by long arrows). Remark 4.2 ( —-+ implies = > ) Let (Sn) and So be in 7£(fi; S). The following hold: (a) If Sn —^ S0, then Sn = > So. (6) If Sn —^ So and if(Sn) is r-tight, then Sn =^=> So. (c) If Sn —^ S0, then Sn =?=> S0. Definition 4.1 obviously extends to a definition of the r- and rp-narrow topologies. In the form given above, the definition of narrow convergence is classical in statis tical decision theory [58, 43]. It merges two completely different classical modes of convergence: Remark 4.3 Let (Sn) and S0 be in 7£(ft;5). The following are obviously equivalent: (a) Sn =^> ¿o in 1Z(ft; S). (b) [fi®6n](A x -)/fi{A) ^ [fi®6o](A x -)/fi(A) in V(S) for every A G A, fx(A) > 0. (c) fs c(x)Sn(-)(dx) -^ fsc(x)S0(')(dx) in £°°(ft;R) for every c G Cb{S,r). Here "—>> " denotes convergence in the topology a(C°°(ft; R),£a (iî; R)). The following example continues the previous Examples 3.2 and 3.11. Example 4.4 Let (il,A,fi) be ([0,1],£([0,1]), Ai) (cf. Example 3.2). As in Exam ple 3.11, let /i G >C1 ([0,1]; R) be arbitrary and extended periodically from [0,1] to all of R. We define / n + i M := fi(2n u>). Then efn = * 60, where S0 G ft([0,l],R) is the constant function given by S0(uj) = Af1 . Here A* G V(H) is the image of Ai under the mapping / i ; i.e., X^(B) := A(/1 ~1 (JB)). TO prove the above convergence statement, let c G C&(R) be arbitrary, and let A be first of the form A = [0,/?] with /? > 0. Then a simple change of variable gives limn^oo fA c(fn)di = fA[fRc(x)So(u))(dx)]duj for A = [0,/?]. By standard methods this can then be extended to all A in A. It follows that Sm in Example 3.11 is equal to the above ¿0, modulo a Ai-null set. The proviso of an exceptional null-set is indispensible, because the narrow limits in 7£(ft; S) are only unique modulo a /¿-null set: 34
- 48. Proposition 4.5 For every 6, 8' in 71(0,] S) the following are equivalent: (a) ¡A[fsc(x)6(uj)(dx)}fi(duj) = ¡A[JS c(x)ó'(uj)(dx)}fi(duj) for every A £ A and ce C0. (b) 6(u>) = 6'(UJ) for a.e. u> in 0,. Theorem 4.6 Suppose that the a-algebra A on O ¿5 countably generated. Then there exists a semimetric dn on 7£(il; S) such that for every (Sn) and S0 in 1Z(íl; S) the following are equivalent: (a) 6n => S0. (b) imndn(Sn,60) = 0. Proof. Let (c¿) enumerate Co of Proposition 2.9, and let (Aj) be the at most countable algebra generating A. Denote qi(A,S) := fA[fsc iô(')(dx ))dv an( ^ define a semimetric on ft(il; S) by dn(6,6') := Ei,j 2"^'|ft(A¿, S) - qi(Ah 8'). To prove (a) =» (6) we note that standard arguments [2, 1.3.11] give limn qi(A, 8n) = qi(A, S0) for every A € A and i. By Proposition 2.9 and Remark 4.3 this implies 6n = > S0. Conversely, (a) =ï (b) is simple. QED Proposition 3.6 and Theorem 3.8 imply the following transfer of the earlier port manteau Theorem 2.4 to the domain of Young measures [10]. T h e o r e m 4.7 (portmanteau t h e o r e m for =>) (i) Let (Sn) and 80 be in 7£(f&; S). The following are equivalent: (a) 6n =£> 60. (6) mn ¡A[fs c(x)6n(u;)(dx)]p(duj) = ¡A[fs c(x)S0(iü)(dx)]p(duj) for every A € A, ceCu(S,p). (c) liminfn Ig(Sn) > Ig(So) for every rp-lower semicontinuous integrand g on il x S such that (ii) Moreover, if (6n) is T-tight, then the above are also equivalent to (d) Sn =^> S0. (e) liminfn Ig{àn) > Ig(õo) for every sequentially r-lower semicontinuous integrand g on Q x S such that Proof. By Remark 4.3 (a) & (b) follows by (a) < £ > (6) in Theorem 2.4. (c) => (6) is obvious: apply (c) to g(w,x) := ±lA(u;)c(x). (a) => (c): By Remark 4.3 vn 4> v0, where vn := [p (g) 6n](Çl x -)/p(il). So by Theorem 2.8 (vn) is rp-tight in V(S): there exists a Tp-inf-compact h' : S —• [0, + oo] such that supn fs h!dvn < +oo. So (Sn) is rp-tight, since fsh'di/n = Ih(Sn)/p(0,) for h(w,x) := hx), Therefore, Theorem 3.8 applies to (Sn). For g as stated, let ¡3 := liminfn Ig(Sn). Then ¡3 = limn/ Ig(ón>) for a suitable subsequence (6n>) and, by Theorem 3.8, we may suppose without loss of 35
- 49. generality that Sn> —^ 6+ for some 6* in 11(0,; S). But in combination with (a) this implies ¿*(u>) = S0(LO) a.e. (Proposition 4.5), so in fact 8n> —^ ¿0. Now ¡3 > Ig(S0) follows from Proposition 3.6. Next, (d) => (a) holds a fortiori and (a) => (e) is proven similarly to (a) => (c), but now r-tightness holds ab initio; let h be as in Definition 3.3. In the remainder of the proof of (a) => (c) we now substitute ge := g + eh, which is a Tp-lower semicontinuous integrand. Letting e — » 0 gives (e). Finally, (e) => (c?) is obvious. QED Results of this kind (but less general) are usually obtained by means of approxima tion procedures for the lower semicontinuous integrands [31, 26, 3, 38, 5, 10, 56, 57], that are completely avoided here. Another difference is that the present approach directly produces results for sequential Young measure convergence. A' Theorem 4.8 (characterization of = > by —> ) (i) Let(Sn) andó0 be inK(0;S). The following are equivalent: (a) Sn = > S0. (b) Every subsequence (¿n>) of (6n) contains a further subsequence (6n») such that 6n» —^ S0. (ii) Moreover, if (Sn) is r-tight, then the above are also equivalent to (c) Sn =h> S0. (d) Every subsequence (6n>) of (Sn) contains a further subsequence (8n») such that Sn» —^ ¿o- In parts (6) and (d) the use of subsequences cannot be replaced by the use of the entire sequence (Sn) itself, simply because a narrowly convergent sequence does not have to A'-converge as a whole [18, Example 3.17]. Corollary 4.9 (i) Let (Sn) and S0 be in 1Z(Q;S). The following are equivalent: (a) Sn => S0 in 1Z(il;S). (b) Sn x en =£» So x Coo in 11(0,; S). (ii) Moreover, if (Sn) is r-tight, then the above are also equivalent to (c) Sn =^ S0 in 11(0; S). (d) Sn x en =^> S0 x Coo in 11(0; 5). Proof, (a) < = > (6) is immediate by Theorem 4.8 and Corollary 3.12. (a) <& (c) is contained in Theorems 4.7 and 4.8. (b) <& (d) is contained in Theorems 4.7 and 4.8, since (Sn x en) is f-tight if and only if (Sn) is r-tight (by compactness of N). QED Transfers of Prohorov's theorem and of the support theorem to Young measure convergence are immediate because of the intermediate results obtained in section 3. The following result is evident by combining Theorem 3.8 and Remark 4.2. See [11] for the topological (i.e., nonsequential) version of this result in precisely the setting of this paper. 36
- 50. Theorem 4.10 (Prohorov's theorem for = > ) (i) Let (8n) in TZ(il;S) be rp-tight. Then there exist a subsequence (6ni) of (8n) and S* G 7£(í); S) such that 8n> = > 8*. (ii) Let (8n) in 1l(iï;S) be r-tight. Then there exist a subsequence (8n>) of (8n) and 8* G 7£(íl; S) such that 8n> =^=> 8*. Example 4.11 We continue with Example 3.5(6). By a(£, E')-tightness of (e/n) we get from Theorem 4.10 that there exist a subsequence (/n/) of (fn) and 6m G 7^(0; E) such that tf , => 8*. (a) We now introduce a function /* G Cl E that is "barycentrically" associated to ¿*, simply by inspecting the consequences of the tightness inequality s :— supn hR{tfn) < +oo that was established there. For hft is a fortiori a cr(E, £")-lower semicontinuous in tegrand, so Theorem 4.7(e) gives hR(8*) < s < H-oo, which implies fs /&/?(u;, x)8*(uj)(dx) < +oo for a.e. UJ. So by the definition of hp. it follows that both 8m(u;)(R(uj)) = 1 and SE x8*{w)(dx) < H-oo for a.e. UJ. By standard facts of Bochner integration it follows that the barycenter /*(u;) := bar ¿*(u;) of the probability measure 8*(ÙJ) is defined for a.e. UJ. Thus, if we set /* := 0 on the exceptional null set, we obtain a func tion /* G £°(íl;E). Finally we notice that, as announced, /* is /i-integrable, i.e., /* G Cl (£l',E). This follows simply from hR(8*) < +oo by use of Jensen's inequality and the inequality hn(uj,x) > x. (6) Suppose that in part (a) one has in addition that (||/n'||) is uniformly integrable in Cl (£l;R). Then fn> A /* G Cl (Q;E) (weak convergence in £1 (Í2;E)). This follows directly from another application of Theorem 4.7(e), namely, to all integrands g of the type g(uj,x) = ± < x,b(uj) >, b G C°°(Q, E')[E}. The latter symbol denotes the set of all scalarly measurable bounded E'-valued functions on Í); it forms the prequotient dual of £1 (fî; E). This yields limn/ Ig(tfn,) = Ig(8*), with Ig(cfn,) = Jg(fn') = in < fw,b(u) > dfi and Ig(6m) = ¡Q < f*,b(u>) > dp. Part (6) in the above example implies that fn A f0 in Example 4.4, where /0 is the constant function given by fo(u>) := bar 8O(UJ) = JnfidXi (apply [35, 11.12]). Concate nation of Theorem 3.13 and Theorem 4.8 gives immediately the following result: Theorem 4.12 (support theorem for = > ) (i) Let 8n = ^ 80 in 7£(i);£). Then Tp-supp 8O(LJ) C rp-Lsnr/7-supp 8n(uj) for a.e. UJ in il. (ii) Moreover, if (8n) is r-tight, then 8n =^> 80 in 7£(f); S) and ¿o(u;)(T-seq-cl r-Lsnr-supp 8n(uj)) = 1, r-supp 8Q(UJ) C r-cl r-Lsnr-supp 8n(uj) for a.e. UJ in il. The following so-called lower closure theorem for Young measures forms a combi nation of the main relative compactness, lower semicontinuity and support results of the present section. Let (D,do) be a metric space. 37
- 51. Theorem 4.13 (fundamental lower closure theorem) Let (6n) in K(iï;S) be r- tight and let dn —• do in C°(Q,; D) (convergence in measure). Then there exist a sub sequence (Snt) of (Sn) and 6* in 7£(0; S) such that for every sequentially f -sequentially lower semicontinuous integrand £ on Q x S x D) such that (4.1) More precisely, we have causing 6* to be supported as follows Here {£ < —a}w > n stands for the set of all x G S for which £(UJ, x, dn(u>)) < —a. Proof. Theorem 3.8 and well-known facts about convergence in measure [28, The orem 20.5] imply existence of a subsequence (6n',dn>) of (6n,dn) and existence of 6* G 7Z(T;S) such that 6n> —^ 6+ in 7£(í); S) and dr)(dnt(u>),d0(uj)) —> 0 for a.e. LO. A fortiori this gives 8n> =^=> ¿* (by Remark 4.2). By Theorem 4.12 this gives the desired pointwise support property for 6*. By Corollary 4.9, we also have 6nt =^> 6* in 1Z(ft; S), with Sn := 6n x en and 6* := ó* x e^ Without loss of generality we discard renumbering and pretend that (nf ) enumerates all the numbers in N. For £ as stated we observe that for each n' G N the following identity holds and it continues to hold for n' — oo if we write d^ := d0 and 6^ := 6*. Here gt(w, (x,k)) := £(tu,x,dk(w)) defines a f-lower semicontinuous integrand g¿ on Q, x S (modulo an insignificant null set). Note in particular that for k = oo lower semicon- tinuity of ge(w, •) at (#,oo) follows from dn>(uj) —> d0(u>) and lower semicontinuity of ¿(u;,-,-) at (x,do(üj)). Thus, the desired inequality is contained in liminfn/ Igt(8ni) > Igt(8*), a result that follows by applying Theorem 4.7 to g¿ (observe here that (4.1) coincides with (3.1) for g = ge). QED Remark 4.14 Let h be the nonnegative, sequentially r-inf-compact integrand h on ClxS that corresponds as in Definition 3.3 to the r-tight sequence (Sn) in Theorem4-13; i.e., with s := supn Ih(ôn) < +oo. Then the uniform integrability condition (4-1) applies whenever the integrand £ has the following growth property with respect to h: for every e > 0 there exists <¡>e G £X (H; R) such that for every n G N £~(uj,x,dn(uj)) < ch(w,x) + <¡>t(u) on O x xS. Indeed, we can observe that the set {£ < —a}w,n in (4-1) is contained in the union of {(j)e < eh} and {(¡>e > a/2}, which gives s'(a) < 3es + f{4>e>a/2} fa ^Pi whence sf (a) —* 0 for a —» oo, as claimed. 38
- 52. 5 Some applications to lower closure and dense- ness We illustrate the power of the above apparatus by some applications to a variety of problems; we refer to [18, 22] for more extensive expositions. As our first application, we derive an extension of the so-called fundamental theo rem for Young measures in [25]. Here L is a locally compact space that is countable at infinity; its usual Alexandrov compactification is denoted by L := LU{oo}. The space L is metrizable, and its metric is denoted by d. On L we use the natural restriction of </, and denote it by d. Let CQ(L) be the usual space of continuous functions on L that converge to zero at infinity. Although it could be avoided by the additional introduction of transition subprobabilities (see the comments below), the Alexandrov compactification L of L figures explicitly in the result. Also, below v denotes a cr-finite measure on (Í!, .4). Corollary 5.1 (i) Let (fn) in £°(il;L) and the closed set C C L be such that limn v(f~x (LG)) = 0 for every open G, C C G C L. Then there exist a subsequence (fn>) °f (In) and 6* in 1l(Q; L) such that 6*(u>)(LC) = 0 for a.e. u in Í1 and for every <j> G £a (í2; R) and every c G Co(L). (ii) Moreover, if for that subsequence there exists a sequence (Kr) of compact sets in L such that limr_>oo supn/ U({LO G fi : fn'{u) $ Kr} = 0 then ¿*(u;)({oo}) = 0 for a.e. u in Í7 and for every A G A, <> G CX (A', R) and c G C(L) for which (lJ4c(/n/)) is relatively weakly compact in £1 (i4;R). In [25] both L and H are Euclidean, and the AVs are closed balls around the origin with radius r. As was done in [25], the result could be equivalently restated in terms of the transition swôprobability 6'm from (i),^4) into (!/,#(!/)), defined by obvious restriction to Z/, i.e., SI(UJ)(B) := 6*(u>)(B U {oo}), B G B(L). In this connection the tightness condition in part (ii) guarantees that 6* is an authentic transition probability (Young measure). Rather than via (¿), part (ii) could also have been derived directly from Theorem 3.8 or 4.13. Proof, (i) By cr-finiteness of z/, there exists a finite measure fi that is equivalent to v. Let ^ be a version of the Radon-Nikodym density dis/dp. Now (¿>n), defined by Sn := tfn G 7£(fi, ;¿), is trivially tight by compactness of L (set h = 0). By Theorem 3.8 or 4.13 (with S := L, p := c?), there exist a subsequence (fn>) of (/n) and 6* in H(Q; L) for which e/n, = > 6* (and even tfn, —^ 6*). Every c G C0(L) has 39
- 53. a canonical extension c G Cb(S) by setting c(oo) = 0. Now (¡><j) is //-integrable for any </> G Cl (Çl,A, J/;R), and Theorem 4.7(c) (or 4.13) can be applied to g : il x L — > R given by g(u,x) := ±</>(L>)<¡>(U)C(X). This gives the desired equality, because of the identity ¡Q ¿(J) fL c(x)6*(-)(dx)dfi = ¡Q<f>fLc(x)6*(-)(dx)di/. Next, let C be as stated. For any i G N the set F¿, consisting of all x G L with ¿/-dist(x, C) < ¿- 1 , is closed in Z. Note already that nt F, = C, by the given Td-closedness of C in L. Further, F{ := F{ U {00} is closed in L. Set gi(u>,x) := <^>(o;)l5vp,.(a;). This defines a nonnegative lower semicontinuous integrand p, on il x L. Hence, 4(¿„) < # := liminfn, 4(£/ n ,) by Theorem 4.7(c). By SFi = LF¡, the definitions of g{ and e/n, give 4(e/ n ,) = i/(/-/(IFt )). So # = liminfn/ */(/-/(LF¿)) < Í/(/~/1 (XG¿)), where G¿, Gt C Ft , is the r^-open set of all x G L with c?-dist(x, C) < i~l . Since G¿ D C, the hypotheses imply 0 = /?,- > Igx{à*) = /nÃ*(-)(LF,-)di/. Hence 6*(u)(LC) = 0 z/-a.e. because of fltF¿ = C, which was demonstrated above. (¿z) The additional condition is a tightness condition for (c/n), when viewed as a subset of 7Z(Q; L) (take I = Kr in Remark 3.4). Hence, there is a rp-inf-compact integrand h on Q x L with supn Ih(ôn) < +00. Now define the inf-compact integrand h on Q x L by h(u;,x) := h(u>,x) if x G X and Ã(u;,oo) := +00. Then /¿(Ó*) < liminfn//^(e/n/) < +00 by Theorem 4.7(c). Hence, £*(-)({°°}) = 0 /¿-a.e., whence z/-a.e. Finally, for any A G -4 with ^(A) < +00 Theorem 4.7(c) applies to g(w,x) := ±IA(W)<I>(W)<J>(W)C(X). This gives the desired limit statement. If v(A) = +00 and A is as stated, there exists a sequence (Aj) of subsets of A with finite i/-measure, with Aj | A. The previous result applies to each of the Aj and the weak relative compactness hypothesis implies uniform <r-additivity [30], so also in this case the desired limit statement follows. QED Next, let E and F be separable Banach spaces, each equipped with a locally convex Hausdorff topology, respectively denoted by TE and r/r, that is not weaker than the weak topology and not stronger than the norm topology. Let (D^do) be a metric space. Functions that are "barycentrically" associated to Young measures can play a special role in lower closure and existence results. This is demonstrated by our proof of the following result. Theorem 5.2 Letdn —> d0 inC°(il]D) (convergence in measure), en A eo ¿n£1 (0;£') (weak convergence), and let (fn) in £1 (i);F) satisfy supn fQ fnFdfi < +00. Suppose that there exist TE- and TF-ball-compact multifunctions RE ' ft — > 2E and Rp : Q, —• 2F such that {(en(u))Jn(u>)) : n G N} C RE{u>) x RF{u)/i-&.e. Then there exist a subsequence (dn>, en/,/n/) of(dn,enyfn) and f* G £1 (Í2;F) such that for every sequentially TEXTFX Try-lower semicontinuous integrand Í on fix (Ex Fx D) such that the following hold: 40
- 54. £{w, -, -,do(u>)) is convex on E x F for a.e. UJ. Moreover, the functions e$ and /* can be localized as follows: 2 (co(c«;),/*(a;)) G cl co-w-Lsn{(en(u;),/n(u;))} for a.e. u> in ÍL Observe, as was already done following Example 3.5, that the ball-compactness condi tion involving RE and Rp is automatically satisfied in case the Banach spaces E and F are reflexive. Proof. To apply Theorem 4.13 we set S := EXF,T := TEXTF and Sn := C(Cn,/n). Then S is completely regular (by the Hahn-Banach theorem) and Suslin. Note that (||en||) in £a (fi; R) is uniformly integrable by [30, Theorem 1] and [45, Proposition II.5.2]. In particular, this implies supn Jn ||(en ,/n )||s dfi < +oo. This proves that (Sn) is r-tight, in view of Example 3.5(6). We can now apply Theorem 4.13: let the subsequence (Sn>,dni) of (6n,dn) and 6* in 7£(í); S) be as guaranteed by that theorem, i.e., with 6ni = ^ 6+ (and even 6n> —^ 6*). Then it is elementary by Definition 4.1 that, U E- marginally", e6n/ ^ ¿f and, "F-marginally", efn, =^> if. Hereof (a;) := i,(w)(.xf), etc. Then ^-marginally Example 4.11(6) applies, which gives that bar Sf = e0 a.e. Also, F-marginally Example 4.11(a) applies, giving the existence of /* G £1 (Q; F) such that /* = bar i f a.e. (note that r#- and rp -ball-compactness imply a(E^E')- and cr(F, F')-ball-compactness respectively). Recombining the above two marginal cases, we find bar 6+ = (e0,/*) a.e. (note that barycenters decompose marginally). We now finish the proof. For an integrand £ of the stated variety Theorem 4.13 gives where ¡3 := liminfn/ /¿¡^(u;,en/(o;),/n/(a;),dn/(a;))//(du;). In the inner integral of the above inequality the convexity of £(UJ, -, -, d0(u>)) gives for a.e. u?, by Jensen's inequality and our previous identity bar 6+ = (eo,/*) a.e. The desired inequality thus follows. QED The above lower closure result "with convexity" is quite general: it further extends the results in [5, 8], which in turn already generalize several lower closure results in the literature, including those for orientor fields (cf. [33]). See [15] for another de velopment, not covered by the above result. Results of this kind are very useful in the existence theory for optimal control and optimal growth theory; e.g., see [33, 15]. Recently, similar-spirited versions that employ quasi-convexity in the sense of Morrey have been given in [42, 52] (these have for en the gradient function of dn and depend on a characterization of so-called gradient Young measures [40, 48]). Corollaries of 2 In case E and F are finite-dimensional one may replace here "cl co" by "co". 41
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- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: La Chèvre d'Or Author: Paul Arène Release date: September 19, 2013 [eBook #43767] Most recently updated: October 23, 2024 Language: French Credits: Produced by Claudine Corbasson, Hans Pieterse and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/Canadian Libraries) *** START OF THE PROJECT GUTENBERG EBOOK LA CHÈVRE D'OR ***
- 61. Au lecteur
- 63. La Chèvre d'Or Ê
- 64. DU MÊME AUTEUR PETITE BIBLIOTHÈQUE LITTÉRAIRE Jean des Figues.—Le Tor d'Entrays.—Le Clos des Ames.—La Mort de Pan.—Le Canot des six capitaines. 1 vol. avec portrait 6 fr. Édition in-18 à 3 fr. 50 Vingt jours en Tunisie 1 vol. Tous droits réservés. P A U L A R È N E La Chèvre d'Or
- 65. P A R I S A L P H O N S E L E M E R R E , É D I T E U R 23-31, PASSAGE CHOISEUL, 23-31 M DCCC XCIII A U D O C T E U R J E A N - M A RT I N C H A R C OT En souvenir de nos voyages au joyeux pays de Provence, permettez-moi, cher Maître et cher Ami, de vous dédier La Chèvre d'Or. Ce petit roman romanesque ne parle pas de névrose. Peut- être vous plaira-t-il à cause de cela.
- 66. P. A. La Chèvre d'Or
- 67. I LETTRE D'ENVOI Ris, ne te gêne point, ami très cher, ô philosophe! Je te vois d'ici lisant ces lignes au fond du fastueux cabinet encombré de la dépouille des âges où, parmi les tableaux anciens, les émaux, les tapisseries, pareil à un Faust qui serait bibelotier, tu passes au creuset de la science moderne ce que l'humanité gardait encore de mystères, et uses tes jours, poussé par je ne sais quel contradictoire et douloureux besoin de vérité, à réduire en vaine fumée les illusions de ce passé dont le reflet pourtant reste ta seule joie; je te vois d'ici, et je devine la compatissante ironie qui, durant une minute, va éclairer ton numismatique profil. Tel que tu me connais, devenu douteur par raison, guéri des beaux enthousiasmes et déshabitué de l'espérance, je suis très sérieusement occupé à la recherche d'un trésor. Oui! ici, en Provence, dans un pays tout de lumière et de belle réalité, aux horizons jamais voilés, aux nuits claires et sans fantômes, je rêve ainsi, éveillé, le plus merveilleux des rêves. Folie! vas-tu dire. Rassure-toi. Bientôt ta sagesse reconnaîtra qu'il me faudrait être fou pour renoncer à ma folie. Car le trésor en question est un trésor réel, palpable, depuis plus de mille ans enfoui, un vrai trésor en or et qui n'a rien de chimérique. Bien que comparable aux amoncellements de joyaux précieux et de
- 68. frissonnantes pierreries dont l'imagination populaire s'éblouissait au temps des Mille et Une Nuits, aucun génie, aucun monstre ne le garde et bientôt il m'appartiendra. Comment?... Laisse-m'en le secret une semaine encore. Du reste j'avais, à ton intention, jeté sur le papier, d'abord pour occuper mes loisirs, plus tard pour amuser mon impatience, le détail exactement noté de mes sensations et de mes aventures depuis le jour de nos adieux. Tu recevras le paquet en même temps que cette lettre. Tout un petit roman dont les circonstances ont seules tissé la trame et où ma volonté ne fut pour rien. Il n'y est question de trésor qu'assez tard. Je t'enverrai bientôt la suite et tu pourras ainsi t'associer aux émotions que je traverse. En attendant, montre-toi indulgent à ma chimère. Pour te prouver que je suis lucide et que la manie des grandeurs ne m'a pas troublé le cerveau, je te jure qu'avant un mois, à Paris, je rirai avec toi et plus fort que toi de mes déconvenues si, au réveil, sous le dernier coup de pioche, je ne trouve, comme dans les contes, à la place du Colchos et de la Golconde espérés qu'un coffre vermoulu, des cailloux et des feuilles sèches.
- 69. II EN VOYAGE Me voici loin, résumons-nous! Le bilan est simple: des amours ou soi-disant tels qui ne m'ont pas donné le bonheur; des travaux impatients qui ne m'ont pas donné la gloire; des amitiés, la tienne exceptée, qui m'ont toutes, en s'évaporant, laissé ce froid au cœur mêlé de sourde colère que provoque l'humiliation de se savoir dupe. Bref! je me retrouve de même qu'au début, avec en moins la foi dans l'avenir et ce don précieux d'être trompé qui, seul, fait la vie supportable. Je ne rappelle que pour mémoire une fortune fort ébréchée sans même que je puisse me donner l'excuse de quelque honorable folie. J'avais très distinct le sentiment de cela, il y a un instant, dans l'éternelle chambre d'hôtel banale et triste, en écoutant l'horloge de la ville sonner. Par une rencontre qui n'a rien de singulier, cette horloge, au milieu de la nuit, sonnait l'heure de ma naissance, cependant qu'à défaut de calendrier, un bouquet d'anniversaire, envoi d'une trop peu oublieuse amie, me disait avec une cruelle douceur le chiffre de mes quarante ans... Ne serait-ce point la cloche d'argent du palais d'Avignon, au même tintement grêle et clair, qui ne sonnait qu'à la mort des papes?
- 70. Il me semble qu'en moi quelque chose vient de mourir. A quoi me résoudre? M'établir pessimiste? Non pas, certes! J'aurais trop peur de ta bien portante raillerie. Après tout, je ne suis plus riche: mais il me reste de quoi vivre libre. Je ne suis plus jeune: mais il y a encore une dizaine de belles années entre l'homme qui m'apparaît dans cette glace et un vieillard. Il est trop tard pour songer à la gloire: mais le travail, même sans gloire, a ses nobles joies. Et, puisque je n'eus pas le génie d'être créateur, peut-être qu'un effort dans l'ordre scientifique, une série de recherches, établies nettement et courageusement poursuivies, me débarrasseront des désespérantes hésitations qui, si souvent, m'ont laissé tomber l'outil des mains à mi-tâche devant des entreprises trop purement imaginatives pour ne pas, à certains moments douloureux, apparaître creuses et vaines au raisonneur et au timide que le hasard a fait de moi. Après avoir cherché, réfléchi, je me suis donc fixé une besogne selon mon courage et mes goûts. Tu sais, s'il m'est permis d'employer une expression que tu affectionnes et que as même, je crois, un peu inventée, quel enragé traditioniste je suis. En exil au milieu du monde moderne, j'ai cette infirmité qu'aucune chose ne m'intéresse si je n'y retrouve le fil d'or qui la rattache au passé. Mon sentiment, d'ailleurs, peut se défendre; l'avenir nous étant fermé, revivre le passé reste encore le seul moyen qui s'offre à nous d'allonger intelligemment nos quelques années d'existence. Tu sais aussi, pour m'avoir souvent plaisanté sur un vague atavisme barbaresque que ton érudition moqueuse me prêtait, tu sais quel faible j'eus toujours pour les souvenirs de la civilisation arabe.
- 71. Dans ce beau pays où, par la langue et par la race, au-dessus du vieux tuf ligure, tant de peuples, Phéniciens, Phocéens, Latins, ont laissé leur marque, les derniers venus, les Arabes seuls m'intéressent. Plus que la Grecque qui, avec ses yeux gris-bleu s'encadrant de longs sourcils noirs, évoque la vision de quelque Cypris paysanne, plus que la Romaine dont souvent tu admiras les fières pâleurs patriciennes, me plaît, rencontrée au détour d'un sentier, la souple et fine Sarrasine, aux lèvres rouges, au teint d'ambre. Et tandis que d'autres sentent leur cœur battre à la trouvaille d'un fragment d'urne antique ou d'une main de déesse que le soleil a dorée, je ne fus jamais tant ému qu'un jour, dans Nîmes, près des bains de Diane, dont les vieilles pierres disparaissaient sous un écroulement de rose, en foulant, parmi les débris, le plafond de marbre fouillé et gaufré que les envahisseurs venus d'Afrique par l'Espagne ajoutèrent ingénument aux ornements ioniens du temple des nymphes. On accueillit en amis, chez nous, ces chevaleresques aventuriers qui, au milieu du dur moyen-âge, nous apportaient, vêtus de soie, la grâce et les arts d'Orient. Quand les Arabes vaincus se réembarquèrent, la Provence entière pleura comme pleurait Blanche de Simiane au départ de son bel émir. J'avais entrepris autrefois sur ce sujet un travail, hélas! interrompu trop vite, et retrouve même fort à propos un carnet jauni dont bien des pages sont restées blanches. Je ferai revivre, en les complétant, ces notes longtemps oubliées. Je recommencerai mes longues courses sous ce ciel pareil au ciel d'Orient, à travers ces rocs mi-africains qui portent le palmier et la figue de Barbarie, le long de ces calanques bleues propices au débarquement, de ces plages où, dans le sable blond, s'enfonçait la proue des tartanes. Heureux le soir et n'ayant pas perdu ma journée, si je découvre quelque nom de famille ou de lieu dont la consonance dise l'origine, si j'aperçois au soleil couchant, près de la mer, sur une cime,
- 72. quelque village blanc, avec une vieille tour sarrasine gardant encore ses créneaux et l'amorce de ses moucharabis. Dans ce pays hospitalier, indulgent aux mauvais chasseurs, un fusil jeté sur l'épaule me donnera l'accès auprès des paysans. La mission, gratuite d'ailleurs et peu déterminée, que ton amitié, à tout hasard, m'avait obtenue du ministère, me fera bien accueillir des savants locaux, des curés, des instituteurs, et me permettra de fouiller les vieux cahiers de tailles, les cadastres, les résidus d'archives. Et, après un mois ou deux de cette érudition en plein air, j'espère te rapporter sinon d'importantes découvertes, du moins un ami solide et bronzé à la place du Parisien ultra-nerveux que tu as envoyé se refaire l'esprit et le corps au soleil.
- 73. III LA PETITE CAMARGUE Mais avant d'entrer en campagne, avant de mettre à exécution tous ces beaux projets, j'aurais besoin de me recueillir quelques jours. Si j'allais demander l'hospitalité à patron Ruf? Il vit sans doute encore. Nous sommes liés depuis quatre ans, et voici comment je fis sa connaissance. Je voyageais, suivant la côte de Marseille à Nice, quand un soir, pas bien loin d'ici, aux environs de l'Estérel, mon attention fut attirée par une demeure rustique dont la singularité m'intéressa. C'était, au pied d'un rocher à pic, une de ces cabanes basses spéciales au delta du Rhône, faites de terre battue et de roseaux, et d'une physionomie si caractéristique avec leur toit blanc de chaux, relevé en corne. Le rocher, évidemment, plongeait autrefois dans la mer; mais l'amoncellement de sables rejetés là par les courants, l'alluvion d'une petite rivière dont l'embouchure paresseuse s'étale en dormantes lagunes avaient peu à peu fait de la baie primitive une étendue de limon saumâtre coupée çà et là de flaques d'eau où poussent des herbes marines, quelques joncs et des tamaris. Trouver ainsi, en pleine Provence levantine, une minuscule Camargue et sa cabane de gardien avait déjà de quoi me surprendre; mais mon étonnement fut au comble quand j'aperçus,
- 74. raccommodant des filets devant la porte, une femme vêtue du costume rhodanien. A mon approche, l'homme sortit. Je le saluai d'un «bien le bonjour!» Au bout d'un moment nous nous trouvions les meilleurs amis du monde. Ruf Ganteaume, et plus usuellement patron Ruf, compromis en 1851 pour avoir, avec son bateau, facilité la fuite de quelques soldats de la résistance, s'en était tiré, ma foi! à bon compte, évitant Cayenne et Lambessa, par un internement aux environs d'Arles. Plus heureux que d'autres, en sa qualité de pêcheur, il put gagner sa vie sur le fleuve, se maria et revint au pays après l'amnistie, avec sa femme, née Tardif, des Tardif de Fourques, et qu'il continuait à appeler Tardive. Ruf et Tardive avaient un fils qu'ils voulurent me présenter. On cria: «Ganteaume! Ganteaume!» Je m'attendais à quelque solide gaillard déjà tanné par le soleil et la mer; je vis sortir d'une touffe de tamaris un tard-venu de dix ans, les cheveux ébouriffés, l'air sauvage, tenant par les pattes une grenouille qu'il venait de capturer. C'était M. l'Aîné, porteur du nom, c'était Ganteaume. Je parvins à apprivoiser Ganteaume, et vécus chez ces braves gens toute une semaine. J'avais promis de leur donner de mes nouvelles. Je ne l'ai point fait. Me reconnaîtront-ils après quatre ans?... Ils m'ont reconnu, et j'ai trouvé toutes choses en état. Une cabane toujours neuve; car Ruf, à chaque automne, en renouvelle la toiture de roseaux, et Tardive, tous les samedis, Ganteaume tenant le seau où flotte la chaux délayée, rebadigeonne crête et murs, suivant la coutume du pays d'Arles. Comme changement, quelques rides sur la face incrustée de sel du patron, et quelques fils d'argent dans les bandeaux grecs de
- 75. Tardive. Ganteaume, poussé vite, est devenu un vaillant garçonnet aux cheveux frisottants de petit blond qui brunira. Ganteaume ne pêche plus aux grenouilles. Quand il ne va pas à la mer, il monte Arlatan, un étalon camarguais, blanc comme la craie, vif comme la poudre, que son père, avec le harnachement en crin tressé, les étriers pleins, la haute selle, ramena de Fourques où l'avait appelé un héritage. Mon installation est bientôt prête. Ganteaume, qui couchera à côté de ses parents, me cède sa chambre; il me semble qu'elle m'attendait. En l'honneur de mon arrivée, on a dîné d'une bouillabaisse pêchée par patron Ruf lui-même et servie, suivant l'usage, sur une écorce de liège oblongue creusée légèrement, pareille à un bouclier barbare. Nous avions chacun pour assiette une moitié de nacre, moules gigantesques aux reflets d'argent et d'acajou que les barques, à grand effort, d'un câble noué en nœud coulant, arrachent dans les récifs du golfe. A part ce détail tout local des assiettes et du plat, j'aurais pu, avec cet horizon d'eaux miroitantes, de tamaris en dentelle sur l'or du couchant, et le clairin d'Arlatan qui tintait, me croire au bord du Vaccarès, dans quelque coin perdu, entre la tour Saint-Louis et les Saintes. Derrière les dunes, la vague chantait. Jusqu'à minuit, Tardive, belle d'humble orgueil, me fit l'éloge de son bonheur. Ganteaume sommeillait. Patron Ruf fumait sans rien dire. Et j'admirais cet inconscient poète qui, pour que sa femme se sentît heureuse et l'aimât, sur un peu de terre amoncelée par l'eau d'un ruisseau, lui avait refait une patrie.
- 76. IV PATRON RUF Patron Ruf, en réalité, vit de sa pêche que Tardive, montée sur Arlatan, va deux ou trois fois par semaine vendre à la ville. Mais son orgueil est d'être corailleur. Ne devient pas corailleur qui veut! Le titre se transmet de père en fils, et les membres de la corporation, une fois reçus, jurent le secret. Un triste métier, paraît-il, que celui de mousse apprenti. Patron Ruf a passé par là, restant des journées entières au fond du bateau —pendant que l'équipage, avant de promener le filet-drague dans les hauts-fonds, s'orientait, pour reconnaître les endroits propices, sur quelque rocher remarqué, quelque ensignadou de la côte—et ne respirant guère que le soir, quand, la journée finie, le bateau amarré, il s'agissait de chercher de l'eau, de ramasser du bois et de faire la bouillabaisse. A seize ans, patron Ruf avait été initié. Et maintenant encore, dès que les mois d'été arrivent, le diable ne l'empêcherait pas d'aller rejoindre la flottille des Confrères au cap d'Antibes. Expéditions mystérieuses où l'on emporte deux, trois jours de vivres, où l'on feint d'embarquer pour Gênes, la Corse, la Sardaigne, bien qu'en somme on ne perde guère la terre de vue.
- 77. Juin approchant, patron Ruf parle de partir, d'emmener cette fois Ganteaume. Mais Tardive gardera Ganteaume, et c'est là leur seule querelle. En attendant, patron Ruf m'a pris pour second. Tous les matins nous filons au large jeter le gangui ou bien tendre les palangrottes. Hier, la mer est devenue grosse subitement. Un peu de mistral soufflait! nous avons dû, au retour, tirer des bordées. Patron Ruf tenait la barre et ne parlait pas. Ganteaume courait pieds nus sur le plat-bord, tout entier à sa voile et à ses cordages. Et tandis que les grandes lames, lentes et lourdes, se déroulaient sous le soleil pareilles à du plomb fondu, je m'amusais, passager inutile, à regarder la côte aride, les collines échelonnées montant ou s'abaissant les unes derrière les autres selon que la bordée nous rapprochait de la rive ou bien nous ramenait au large. A la cime d'un pic, dans le soleil, une tache blanche brillait. Je demandai:—«Est-ce un village?—Le Puget..., répondit patron Ruf sans lâcher sa pipe.—Le Puget-Maure!» ajouta Ganteaume. L'aspect du lieu, ce nom sarrasin, surexcitaient ma curiosité savante. J'aurais voulu d'autres détails. Mais patron Ruf, furieux d'un coup de barre donné à faux, s'obstinait dans sa taciturnité; malgré mon impatience, je dus me résigner et attendre que la belle humeur lui revînt avec le beau temps. Aujourd'hui le vent a augmenté. Au cagnard, entre deux buttes de sable tiède où le vif soleil des jours de mistral allume des paillettes, nous causons, patron Ruf et moi, tandis que là-bas Tardive cuisine et que Ganteaume vagabonde sur la plage ramassant, pour me les montrer, des coquilles, des os de seiche, des pierres ponces, et les épis d'algue feutrés en boules brunes que rejette au milieu de flocons d'écume la grande colère de la mer.
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