![viii Contents
4.8 Measurable Function and Measurable Set 94
4.9 Convergence of Measurable Functions 109
4.10 Product Integration and Fubini’s Theorem 119
5 Probability Space 138
5.1 Random Variable 138
5.2 Probability Distribution on Metric Space 151
5.3 Weak Convergence of Distributions 155
5.4 Probability Density Function and Distribution Function 164
5.5 Skorokhod Representation 169
5.6 Independence and Conditional Expectation 182
5.7 Normal Distribution 189
5.8 Characteristic Function 201
5.9 Central Limit Theorem 220
PART III STOCHASTIC PROCESS 225
6 Random Field and Stochastic Process 227
6.1 Random Field and Finite Joint Distributions 227
6.2 Consistent Family of f.j.d.’s 231
6.3 Daniell–Kolmogorov Extension 241
6.4 Daniell–Kolmogorov–Skorokhod Extension 262
7 Measurable Random Field 275
7.1 Measurable r.f. That Is Continuous in Probability 275
7.2 Measurable Gaussian Random Field 290
8 Martingale 299
8.1 Filtration and Stopping Time 299
8.2 Martingale 306
8.3 Convexity and Martingale Convergence 314
8.4 Strong Law of Large Numbers 323
9 a.u. Continuous Process 331
9.1 Extension from Dyadic Rational Parameters to Real
Parameters 333
9.2 C-Regular Family of f.j.d.’s and C-Regular Process 337
9.3 a.u. Hoelder Process 346
9.4 Brownian Motion 353
9.5 a.u. Continuous Gaussian Process 362
10 a.u. Càdlàg Process 374
10.1 Càdlàg Function 375
10.2 Skorokhod Space D[0,1] of Càdlàg Functions 384
10.3 a.u. Càdlàg Process 405
10.4 D-Regular Family of f.j.d.’s and D-Regular Process 409](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-14-320.jpg)
![Nomenclature
≡........................................... by definition equal to, 8
R........................................... set of real numbers, 8
decld...................................... Euclidean metric, 8
a ∨ b..................................... max(a,b),8
a ∧ b..................................... min(a,b),8
a+......................................... max(a,0),8
a−......................................... min(a,0), 8
A ∪ B.................................... union of sets A and B, 8
A ∩ B,AB.............................intersection of sets A and B, 8
[a]1........................................an integer [a]1 ∈ (a,a + 2) for given a ∈ R, 9
X|A....................................... restriction of function X on a set to a subset A, 9
X ◦ X,X (X)....................... composite of functions X and X, 10
(X ≤ a)................................. {ω ∈ domain(X) : X(ω) ≤ a}, 11
X(·,ω ).... ............................ function of first variable, given value of second variable for a function X
of two variables, 12
T ∗(Y) ≡ T (·,Y)................... dual function of Y relative to a certain mapping T , 12
(S,d) .....................................metric space, with metric d on set S, 12
x y .................................... d(x,y) 0, where x,y are in some metric space, 13
Jc ..........................................metric complement of subset J in a metric space, 12
.......................................... direct product of functions or sets, 14
Cu(S,d),Cu(S).....................space of uniformly continuous real-valued functions on metric space
(S,d), 14
Cub(S,d),Cub(S)................. subspace of Cu(S,d) whose members are bounded, 14
C0(S,d), C0(S).....................subspace of Cu(S,d) whose members vanish at infinity, 15
C(S,d),C(S).........................subspace of Cu(S,d) whose members have bounded supports, 15
d............................................ 1 ∧ d, 15
O,o........................................bounds for real-valued function, 16
............................................ mark for end of proof or end of definition, 16
ξ............................................ binary approximation of a metric space, 20
ξ ........................................ modulus of local compactness corresponding to ξ, 20
(S,d)..................................... one-point compactification of (S,d), 34
........................................... point at infinity, 34
F|B....................................... {f |B : f ∈ F}, 35
+∞
−∞ X(x)dF(x)..................Riemann–Stieljes integral, 46
1A..........................................indicator of measurable set A, 67
Ac..........................................measure-theoretic complement of measurable set A, 67
♦........................................... ordering between certain real numbers and functions, 73
xi](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-17-320.jpg)
![xii Nomenclature
(G,λ).....................................profile system, 73
(a,b) α............................. the interval (a,b) is bounded in profile by α, 74
(,L,E)................................probability space, 138
E(dω)X(ω)....................... E(X), 139
ρP rob.................................... the probability metric on r.v.’s, 145
L(G)..................................... probability subspace generated by the family G of r.v.’s, 150
J(S,d)...................................set of distributions on complete metric space (S,d), 151
⇒.......................................... weak convergence of distributions, or convergence in distributions of
r.r.v.’s, 155
ρDist,ξ .................................. metric on distributions on a locally compact metric space relative to
binary approximation ξ, 156
FX......................................... P.D.F. induced on R by an r.r.v. X, 165
Sk,ξ .................................... Skorokhod representation of distributions on (S,d), determined by ξ, 170
L|L ....................................... subspace of conditionally integrable r.r.v.’s given the subspace L , 184
L|G........................................subspace of conditionally integrable r.r.v.’s, given L(G), 184
EA.........................................conditional expectation given an event A with positive probability, 184
ϕμ,σ ...................................... multivariate normal p.d.f., 192
μ,σ ..................................... multivariate normal distribution, 193
ϕ0,I ....................................... multivariate standard normal p.d.f., 193
0,I ...................................... multivariate standard normal distribution, 193
........................................... tail of univariate standard normal distribution, 193
ψX.........................................characteristic function of r.v. X with values in Rn, 204
ψJ ......................................... characteristic function of distribution J on Rn, 204
g............................................ Fourier transform of complex-valued function g on Rn, 204
f g......................................Convolution of complex-valued functions f and g on Rn, 204
ρchar ..................................... metric on characteristic functions on Rn, 210
R(Q × ,S)......................... set of r.f.’s with parameter set Q, state space (S,d), and sample space
(,L,E), 227
X|K...................................... restriction of X ∈
R(Q × ,S) to parameter subset K ⊂ Q, 227
δCp,K ....................................modulus of continuity in probability of X|K, 228
δcau,K ................................... modulus of continuity a.u. of X|K, 228
δauc,K ................................... modulus of a.u. continuity of X|K, 228
F(Q,S)................................. set of consistent families of f.j.d.’s with parameter set Q and state space
S, 232
ρMarg,ξ,Q............................ marginal metric for the set
F(Q,S) relative to the binary approximation
ξ, 237
FCp(Q,S)............................. subset of
F(Q,S) whose members are continuous in probability, 238
ρCp,ξ,Q,Q(∞)...................... metric on
FCp(Q,S) relative to dense subset Q∞of parameter metric
space Q, 240
ρP rob,Q................................ probability metric on
R(Q × ,S), 265
ρSup,P rob............................. metric on
FCp(Q,S), 277
RCp(Q × ,S).....................subset of
R(Q×,S) whose members are continuous in probability, 228
RMeas(Q × ,S).................subset of
R(Q × ,S) whose members are measurable, 276
RMeas,Cp(Q × ,S)...........
RMeas(Q × ,S) ∩
RCp(Q × ,S), 276
L............................................filtration in probability space (,L,E), 300
LX........................................ natural filtration of a process X, 300
L+........................................ right-limit extension of filtration L, 301
L(τ)....................................... probability subspace of observables at stopping time τ relative to filtra-
tion L, 302
λ............................................ the special convex function on R, 316
Qm,Qm,
Qm,Q∞,Q∞........certain subsets of dyadic rationals in [0,∞), 332
(
C[0,1],ρ
C[0,1])................... metric space of a.u. continuous processes on [0,1], 334](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-18-320.jpg)
![Nomenclature xiii
Lim.........................................................extension by limit of a process with parameter set Q∞ to
parameter set [0,1], 335
D[0,1]....................................................... set of all a.u. càdlàg processes on [0,1], 406
δaucl.......................................................... modulus of a.u. càdlàg, 406
Dδ(aucl),δ(cp)[0,1]....................................subset of
D[0,1] whose members have moduli δCp, and
δaucl, 406
ρ
D[0,1]...................................................... metric on
D[0,1], 409
(
RDreg(Q∞ × ,S),
ρP rob,Q(∞))......... metric space of D-regular processes, 410
rLim....................................................... extension by right-limit of a process with parameter set Q∞
to parameter set [0,1], 418
βauB......................................................... modulus of a.u. boundedness, 445
δSRCp....................................................... modulus of strong right continuity in probability, 445
τf,a,N (X)................................................ certain first exit times by the process X, 488
T................................................................ a Markov semigroup, 500
δT.............................................................. a modulus of strong continuity of T, 500
αT............................................................. a modulus of smoothness of T, 500
F∗,T
r(1),···,r(m)............................................. a finite joint transition distribution generated by T, 502
(T ,ρT )................................................... metric space of Markov semigroups. 525
V............................................................... a Feller semigroup, 538
δV.............................................................. a modulus of strong continuity of V, 538
αV............................................................. a modulus of smoothness of V, 538
κV..............................................................a modulus of nonexplosion of V, 538
F∗,V
r(1),···,r(m)..............................................a finite joint transition distribution generated by V, 539
((S,d),(,L,E),{Ux,V : x ∈ S})............ Feller process, 543
((Rm,dm),(,L,E),{Bx : x ∈ Rm})...... Brownian motion as a Feller process, 568](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-19-320.jpg)
![1
Introduction
The present work on probability theory is an outgrowth of the constructive analysis
in [Bishop 1967] and [Bishop and Bridges 1985].
Perhaps the simplest explanation of constructive mathematics is by way of
focusing on the following two commonly used theorems. The first, the principle
of finite search, states that, given a finite sequence of 0-or-1 integers, either all
members of the sequence are equal to 0, or there exists a member that is equal
to 1. We use this theorem without hesitation because, given the finite sequence, a
finite search would determine the result.
The second theorem, which we may call the principle of infinite search, states
that, given an infinite sequence of 0-or-1 integers, either all members of the
sequence are equal to 0, or there exists a member that is equal to 1. The name
“infinite search” is perhaps unfair, but it brings into sharp focus the point that
the computational meaning of this theorem is not clear. The theorem is tantamount
to an infinite loop in computer programming without any assurance of termination.
Most mathematicians acknowledge the important distinction between the two
theorems but regard the principle of infinite search as an expedient tool to
prove theorems, with the belief that theorems so proved can then be specialized
to constructive theorems, when necessary. Contrary to this belief, many clas-
sical theorems proved directly or indirectly via the principle of infinite search
are actually equivalent to the latter: as such, they do not have a constructive
proof. Oftentimes, not even the numerical meaning of the theorems in question
is clear.
We believe that, for the constructive formulations and proofs of even the most
abstract theorems, the easiest way is to employ a disciplined and systematic
approach, by using only finite searches and by quantifying mathematical objects
and theorems at each and every step, with natural numbers as a starting point.
The references cited earlier show that this approach is not only possible but also
fruitful.
It should be emphasized that we do not claim that theorems whose proofs
require the principle of infinite search are untrue or incorrect. They are certainly
correct and consistent derivations from commonly accepted axioms. There is
3](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-23-320.jpg)
![Introduction 5
regard Part III of this book, Chapters 6–11, the part on stochastic processes, as an
extension of Kolmogorov’s constructive methods to stochastic processes: Daniell–
Kolmogorov–Skorokhod construction of random fields, measurable random
fields, a.u. continuous processes, a.u. càdlàg processes, martingales, strong
Markov processes, and Feller processes, all with locally compact state spaces.
Such a systematic, constructive, and general treatment of stochastic processes,
we believe, has not previously been attempted.
The purpose of this book is twofold. First, a student with a general mathematics
background at the first-year graduate-school level can use it as an introduction to
probability or to constructive mathematics. Second, an expert in probability can
use it as a reference for further constructive development in his or her own research
specialties.
Part II of this book, Chapters 3–5, is a repackaging and expansion of the mea-
sure theory in [Bishop and Bridges 1985]. It enables us to have a self-contained
probability theory in terms familiar to probabilists.
For expositions of constructive mathematics, see the first chapters of the last
cited reference. See also [Richman 1982] and [Stolzenberg 1970]. We give a
synopsis in Chapter 2, along with basic notations and terminologies for later
reference.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-25-320.jpg)
![Preliminaries 7
2.3 Proof by Contradiction
There is a trivial form of proof by contradiction that is valid and useful in
constructive mathematics. Suppose we have already proved that one of two given
alternatives, A and B, must hold, meaning that we have given a finite method,
that, when unfolded, gives either a proof for A or a proof for B. Suppose subse-
quently we also prove that A is impossible. Then we can conclude that we have
a proof of B; we need only exercise said finite method, and see that the resulting
proof is for B.
2.4 Recognizing Nonconstructive Theorems
Consider the simple theorem “if a is a real number, then a ≤ 0 or 0 a,” which
may be called the principle of excluded middle for real numbers. We can see that
this theorem implies the principle of infinite search by the following argument.
Let (x)i=1,2,... be any given sequence of 0-or-1 integers. Define the real number
a =
∞
i=1 xi2−i. If a ≤ 0, then all members of the given sequence are equal to 0;
if 0 a, then some member is equal to 1. Thus the theorem implies the principle
of infinite search, and therefore cannot have a constructive proof.
Consequently, any theorem that implies this limited principle of excluded
middle cannot have a constructive proof. This observation provides a quick test to
recognize certain theorems as nonconstructive. Then it raises the interesting task
of examining the theorem for constructivization of a part or the whole, or the
task of finding a constructive substitute of the theorem that will serve all future
purposes in its stead.
For the aforementioned principle of excluded middle of real numbers, an
adequate constructive substitute is the theorem “if a is a real number, then,
for arbitrarily small ε 0, we have a ε or 0 a.” Heuristically, this is
a recognition that a general real number a can be computed with arbitrarily small,
but nonzero, error.
2.5 Prior Knowledge
We assume that the reader of this book has familiarity with calculus, real analysis,
and metric spaces, as well as some rudimentary knowledge of complex analysis.
These materials are presented in the first chapters of [Bishop and Bridges 1985].
We will also quote results from typical undergraduate courses in calculus or linear
algebra, with the minimal constructivization wherever needed.
We assume also that the reader has had an introductory course in probability
theory at the level of [Feller I 1971] or [Ross 2003]. The reader should have
no difficulty in switching back and forth between constructive mathematics
and classical mathematics, or at least no more than in switching back and forth
between classical mathematics and computer programming. Indeed, the reader is
urged to read, concurrently with this book if not before delving into it, the many
classical texts in probability.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-27-320.jpg)
![8 Introduction and Preliminaries
2.6 Notations and Conventions
If x,y are mathematical objects, we write x ≡ y to mean “x is defined as y,”
“x, which is defined as y,” “x, which has been defined earlier as y,” or any other
grammatical variation depending on the context.
2.6.1 Numbers
Unless otherwise indicated, N,Q, and R will denote the set of integers, the set
of rational numbers in the decimal or binary system, and the set of real numbers,
respectively. We will also write {1,2, . . .} for the set of positive integers. The set
R is equipped with the Euclidean metric d ≡ decld. Suppose a,b,ai ∈ R for
i = m,m + 1, . . . for some m ∈ N. We will write limi→∞ ai for the limit of the
sequence am,am+1, . . . if it exists, without explicitly referring to m. We will write
a ∨ b,a ∧ b,a+, and a− for max(a,b), min(a,b),a ∨ 0, and a ∧ 0, respectively.
The sum
n
i=m ai ≡ am + · · · + an is understood to be 0 if n m. The product
n
i=m ai ≡ am · · · an is understood to be 1 if n m. Suppose ai ≥ 0 for i = m,
m + 1, . . . We write
∞
i=m ai ∞ if and only if
∞
i=m |ai| ∞, in which
case
∞
i=m ai is taken to be limn→∞
n
i=m ai. In other words, unless otherwise
specified, convergence of a series of real numbers means absolute convergence.
Regarding real numbers, we quote Lemma 2.18 from [Bishop and Bridges 1985],
which will be used, extensively and without further comments, in the present book.
Limited proof by contradiction of an inequality of real numbers. Let x,y be
real numbers such that the assumption x y implies a contradiction. Then x ≤ y.
This lemma remains valid if the relations and ≤ are replaced by and ≥,
respectively.
We note, however, that if the relations and ≤ are replaced by ≥ and
, respectively, then the lemma would not have a constructive proof. Roughly
speaking, the reason is that a constructive proof of x y implies the calculation
of a positive ε 0 such that y − x ε, which is more than a proof of x ≤ y;
the latter requires only a proof that x y is impossible and does not require
the calculation of anything. The reader should ponder on the subtle but important
difference.
2.6.2 Set, Operation, and Function
Set. In general, a set is a collection of objects equipped with an equality relation.
To define a set is to specify how to construct an element of the set, and how to
prove that two elements are equal. A set is also called a family.
A member ω in the collection is called an element of the latter, or, in symbols,
ω ∈ .
The usual set-theoretic notations are used. Let two subsets A and B of a set
be given. We will write A∪B for the union, and A∩B or AB for the intersection.
We write A ⊂ B if each member ω of A is a member of B. We write A ⊃ B for](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-28-320.jpg)
![Preliminaries 9
B ⊂ A. The set-theoretic complement of a subset A of the set is defined as the
set {ω ∈ : ω ∈ A implies a contradiction}. We write ω A if ω ∈ A implies a
contradiction.
Nonempty set. A set is said to be nonempty if we can construct some element
ω ∈ .
Empty set. A set is said to be empty if it is impossible to construct an element
ω ∈ . We will let φ denote an empty set.
Operation. Suppose A,B are sets. A finite, step-by-step, method X that
produces an element X(x) ∈ B given any x ∈ A is called an operation from A
to B. The element X(x) need not be unique. Two different applications of the
operation X with the same input element x can produce different outputs. An
example of an operation is [·]1, which assigns to each a ∈ R an integer [a]1 ∈
(a,a + 2). This operation is a substitute of the classical operation [·] and will be
used frequently in the present work.
Function. Suppose , are sets. Suppose X is an operation that, for each ω
in some nonempty subset A of , constructs a unique member X(ω) in . Then
the operation X is called a function from to , or simply a function on .
The subset A is called the domain of X. We then write X : → , and write
domain(X) for the set A. Thus a function X is an operation that has the additional
property that if ω1 = ω2 in domain(X), then X(ω1) = X(ω2) in . To specify
a function X, we need to specify its domain as well as the operation that produces
the image X(ω) from each given member ω of domain(X).
Two functions X,Y are considered equal, X = Y in symbols, if
domain(X) = domain(Y),
and if X(ω) = Y(ω) for each ω ∈ domain(X). When emphasis is needed, this
equality will be referred to as the set-theoretic equality, in contradistinction to
almost everywhere equality, to be defined later.
A function is also called a mapping.
Domain of function on a set need not be the entire set. The nonempty
domain(X) is not required to be the whole set . This will be convenient when
we work with functions defined only almost everywhere, in a sense to be made
precise later in the setting of a measure/integration space. Henceforth, we write
X(ω) only with the implicit or explicit condition that ω ∈ domain(X).
Miscellaneous set notations and function notations. Separately, we some-
times use the expression ω → X(ω) for a function X whose domain is understood.
For example, the expression ω → ω2 stands for the function X : R → R defined
by X(ω) ≡ ω2 for each ω ∈ R.
Let X : → be a function, and let A be a subset of such that A ∩
domain(X) is nonempty. Then the restriction X|A of X to A is defined as the
function from A to with domain(X|A) ≡ A ∩ domain(X) and (X|A)(ω) for
each ω ∈ domain(X|A). The set](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-29-320.jpg)
![12 Introduction and Preliminaries
We make similar definitions when the relation ≤ is replaced by , ≥, , or =.
We say X is nonnegative if X ≥ 0.
Suppose a ∈ R. We will abuse notations and write a also for the constant
function X with domain(X) = and with X(ω) = a for each ω ∈ domain(X).
Regarding one of several variables as a parameter. Let X be a function on
the product set × . Let ω ∈ be such that (ω ,ω ) ∈ domain(X) for some
ω ∈ . Define the function X(ω ,·) on by
domain(X(ω ,·)) ≡ {ω ∈ : (ω ,ω ) ∈ domain(X)},
and by X(ω ,·)(ω ) ≡ X(ω ,ω ) for each ω ∈ domain(X(ω ,·)). Similarly,
let ω ∈ be such that (ω ,ω ) ∈ domain(X) for some ω ∈ . Define the
function X(·,ω ) on by
domain(X(·,ω )) ≡ {ω ∈ : (ω ,ω ) ∈ domain(X)},
and by X(·,ω )(ω ) ≡ X(ω ,ω ) for each ω ∈ domain(X(·,ω )).
More generally, given a function X on the Cartesian product × × · · · ×
(n), for each (ω ,ω , . . . ,ω(n)) ∈ domain(X), we define similarly the functions
X(·,ω ,ω , . . . ,ω(n)), X(ω , ·,ω , . . . ,ω(n)), . . .,X(ω ,ω , . . . ,ω(n−1),·) on the
sets , , . . . ,(n), respectively.
Let M ,M denote the families of all real-valued functions on two sets , ,
respectively, and let L be a subset of M ’. Suppose
T : × L → R (2.6.1)
is a real-valued function. We can define the function
T ∗
: L → M
with
domain(T ∗
) ≡ {Y ∈ L : domain(T (·,Y )) is nonempty}
and by T ∗(Y) ≡ T (·,Y). When there is no risk of confusion, we write T also for
the function T ∗, we write T Y for T (·,Y), and we write
T : L → M
interchangeably with the expression 2.6.1. Thus the duality
T (·,Y)(ω ) ≡ T (ω ,Y) ≡ T (ω ,·)(Y). (2.6.2)
2.6.3 Metric Space
The definitions and notations related to metric spaces in [Bishop and Bridges 1985],
with few exceptions, are familiar to readers of classical texts. A summary of these
definitions and notations follows.
Metric complement. Let (S,d) be a metric space. If J is a subset of S, its metric
complement is the set {x ∈ S : d(x,y) 0 for all y ∈ J}. Unless otherwise
specified, Jc will denote the metric complement of J.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-32-320.jpg)
![Preliminaries 15
each ε 0, there exists a 0 such that |f | ≤ ε for each x ∈ S with d(x,x◦) a.
Write C0(S,d), or simply C0(S), for the space of continuous functions on (S,d)
that vanish at infinity.
Space of real-valued continuous functions with bounded support. A real-
valued function f on S is said to have a subset A ⊂ S as support if x ∈ domain(f )
and |f (x)| 0 together imply x ∈ A. Then we also say that f is supported by
A, or that A supports f . We will write C(S,d), or simply C(S), for the subspace
of Cu(S,d) whose members have bounded supports. In the case where (S,d) is
locally compact, C(S) consists of continuous functions on (S,d) with compact
supports. Summing up,
C(S) ⊂ C0(S) ⊂ Cub(S) ⊂ Cu(S).
Infimum and supremum. Suppose a subset A of R is nonempty. A number
b ∈ R is called a lower bound of A, and A said to bounded from below, if b ≤ a
for each a ∈ A. A lower bound b of A is called the greatest lower bound, or
infimum, of A if b ≥ b for each lower bound b of A. In that case, we write
inf A ≡ b.
Similarly, a number b ∈ R is called an upper bound of A, and A said to be
bounded from above, if b ≥ a for each a ∈ A. An upper bound b of A is called
the least upper bound, or supremum, of A if b ≤ b for each upper bound b of A.
In that case, we write sup A ≡ b.
In contrast to classical mathematics, there is no constructive general proof for
the existence of a infimum for an subset of R that is bounded from below. Existence
needs to be proved before each usage for each special case, much as in the case of
limits.
In that regard, [Bishop and Bridges 1985] prove that if a nonempty subset A
of R is totally bounded, then both inf A and sup A exist. Moreover, suppose f
is a continuous real-valued function on a compact metric space (K,d). Then the
last cited text proves that infK f ≡ inf{f (x) : x ∈ K} and supK f ≡ sup{f (x) :
x ∈ K} exist.
2.6.4 Miscellaneous
Notations for if, only if, etc. The symbols ⇒, ⇐, and ⇔ will in general stand
for “only if,” “if,” and “if and only if,” respectively. An exception will be made
where the symbol ⇒ is used for weak convergence, as defined later. The intended
meaning will be clear from the context.
Capping a metric at 1. If (S,d) is a metric space, we will write
d ≡ 1 ∧ d.
Abbreviation for subsets. We will often write “x,y, . . . ,z ∈ A” as an
abbreviation for “{x,y, . . . ,z} ⊂ A.”
Default notations for numbers. Unless otherwise indicated by the context,
the symbols i,j,k,m,n,p will denote integers, the symbols a,b will denote real](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-35-320.jpg)
![3
Partition of Unity
In the previous chapter, we summarized the basic concepts and theorems about
metric spaces from [Bishop and Bridges 1985]. Locally compact metric space was
introduced. It is a simple and wide-ranging generalization of the real line. With few
exceptions, the metric spaces used in the present book are locally compact.
In the present chapter, we will define and construct binary approximations of a
locally compact metric space (S,d), then define and construct a partition of unity
relative to each binary approximation. Roughly speaking, a binary approximation
is a digitization of (S,d), a generalization of the dyadic rationals that digitize the
space R of real numbers. A partition of unity is then a sequence in C(S,d) that
serves as a basis for the linear space C(S,d) of continuous functions on (S,d)
with compact supports, in the sense that each f ∈ C(S,d) can be approximated
by linear combinations of members in the partition of unity.
A partition of unity provides a countable set of basis functions for the metriza-
tion of probability distributions on the space (S,d). Because of that important role,
we will study binary approximations and partitions of unity in detail in this chapter.
3.1 Abundance of Compact Subsets
First we cite a theorem from [Bishop and Bridges 1985] that guarantees an abun-
dance of compact subsets.
Theorem 3.1.1. Abundance of compact sets. Let f : K → R be a continuous
function on a compact metric space (K,d) with domain(f ) = K. Then, for all
but countably many real numbers α infK f , the set
(f ≤ α) ≡ {x ∈ K : f (x) ≤ α}
is compact.
Proof. See theorem (4.9) in chapter 4 of [Bishop and Bridges 1985].
Classically, the set (f ≤ α) is compact for each α ≥ infK f , without exception.
Such a general theorem would, however, imply the principle of infinite search and
is therefore nonconstructive. Theorem 3.1.1 is sufficient for all our purposes.
17](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-37-320.jpg)
![20 Introduction and Preliminaries
Recall that (i) |A| denotes the number of elements in an arbitrary finite set A;
(ii) a subset A of S is said to be metrically discrete if for each y,z ∈ A, either
y = z or d(y,z) 0; and (iii) a finite subset A of a subset K ⊂ S is called an
ε-approximation of K if for each x ∈ K, there exists y ∈ A with that d(x,y) ≤
ε. Classically, each subset of (S,d) is metrically discrete. Condition (iii) can be
written more succinctly as
K ⊂
x∈A
(d(·,x) ≤ ε).
Definition 3.2.1. Binary approximation and modulus of local compactness.
Let (S,d) be a locally compact metric space, with an arbitrary but fixed reference
point x◦. Let A0 ≡ {x◦} ⊂ A1 ⊂ A2 ⊂ . . . be a sequence of metrically discrete
and finite subsets of S. For each n ≥ 1, write κn ≡ |An|. Suppose
(d(·,x◦) ≤ 2n
) ⊂
x∈A(n)
(d(·,x) ≤ 2−n
) (3.2.1)
and
x∈A(n)
(d(·,x) ≤ 2−n+1
) ⊂ (d(·,x◦) ≤ 2n+1
) (3.2.2)
for each n ≥ 1. Then the sequence ξ ≡ (An)n=1,2,... of subsets is called a binary
approximation for (S,d) relative to x◦, and the sequence of integers
ξ ≡ (κn)n=1,2,... ≡ (|An|)n=1,2,...
is called the modulus of local compactness of (S,d) corresponding to ξ.
Thus a binary approximation is an expanding sequence of 2−n-approximation
for (d(·,x◦) ≤ 2n) as n → ∞. The next proposition shows that the definition is
not vacuous.
First note that
∞
n=1 An is dense in (S,d) in view of relation 3.2.1. In the case
where (S,d) is compact, for n ≥ 1 so large that S = (d(·,x◦) ≤ 2n), relation 3.2.1
says that we need at most κn points to make a 2−n-approximation of S.1
Lemma 3.2.2. Existence of metrically discrete ε-approximations. Let K be a
compact subset of the locally compact metric space (S,d). Let A0 ≡ {x1, . . . ,xn}
be an arbitrary metrically discrete finite subset of K. Let ε 0 be arbitrary. Then
the following conditions hold:
1. There exists a metrically discrete finite subset A of K such that (i) A0 ⊂ A
and (ii) A is an ε-approximation of K.
2. In particular, there exists a metrically discrete finite set A that is an
ε-approximation of K.
1 Incidentally, the number log κn is a bound for Kolmogorov’s 2−n-entropy of the compact met-
ric space (S,d), which represents the informational content in a 2−n-approximation of S. See
[Lorentz 1966] for a definition of ε-entropy.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-40-320.jpg)
![Partition of Unity 29
Thus the function f ⊗f : S ×S → R has Lipschitz constant a +a . Assertion 6
follows.
7. The proof of Assertion 7 is omitted.
The next definitions and propositions embellish proposition 6.15 on page 119
of [Bishop and Bridges 1985].
Definition 3.3.2. ε-Partition of unity. Let A be an arbitrary metrically discrete
and finite subset of a locally compact metric space (S,d). Because the set A is
finite, we can write A = {x1, . . . ,xκ} for some sequence x ≡ (x1, . . . ,xκ), where
x : {1, . . . ,κ} → A is an enumeration of the finite set A. Thus |A| ≡ κ. Let ε 0
be arbitrary. Define, for each k = 1, . . . ,κ, the function
ηk ≡ 1 ∧ (2 − ε−1
d(·,xk))+ ∈ C(S,d) (3.3.2)
and
g+
k ≡ η1 ∨ · · · ∨ ηk ∈ C(S,d). (3.3.3)
In addition, define g+
0 ≡ 0. Also, for each k = 1, . . . ,κ, define
gx(k) ≡ g+
k − g+
k−1. (3.3.4)
Then the subset {gy : y ∈ A} of C(S,d) is called the ε-partition of unity of
(S,d), determined by the enumerated set A. The members of {gy : y ∈ A} are
called the basis functions of the ε-partition of unity.
Proposition 3.3.3. Properties of ε-partition of unity. Let A = {x1, . . . ,xκ} be
an arbitrary metrically discrete and enumerated finite subset of a locally compact
metric space (S,d). Let ε 0 be arbitrary. Let {gx : x ∈ A} be the ε-partition of
unity determined by the enumerated set A. Then the following conditions hold:
1. gx has values in [0,1] and has the set (d(·,x) 2ε) as support, for each
x ∈ A.
2.
x∈A gx ≤ 1 on S.
3.
x∈A gx = 1 on
x∈A(d(·,x) ≤ ε).
4. For each x ∈ A, the functions gx,
y∈A;yx gy, and
y∈A gy have Lipschitz
constant 2ε−1. Here y x means y = xi and x = xj for some i,j ∈ {1, . . . ,κ}
with i j.
Proof. 1. Use the notations in Definition 3.3.2. Let k = 1, . . . ,κ be arbitrary.
Suppose y ∈ S is such that gx(k)(y) 0. By the defining equality 3.3.4, it follows
that g+
k (y) g+
k−1(y). Hence ηk(y) 0 by equality 3.3.3. Equality 3.3.2 then
implies that d(y,xk) 2ε. Thus we see that the function gx(k) has (d(·,xk) 2ε)
as support. In general, gx(k) has values in [0,1], thanks to equalities 3.3.2, 3.3.3,
and 3.3.4. Assertion 1 is proved.
2. Note that
x∈A gx = g+
κ ≡ η1 ∨ · · · ∨ ηκ ≤ 1. Assertion 2 is verified.
3. Suppose y ∈ S is such that d(y,xk) ≤ ε for some k = 1, . . . ,κ. Then
ηk(y) = 1 according to equality 3.3.2. Hence
x∈A gx(y) = g+
k (y) = 1 by
equality 3.3.3. Assertion 3 is proved.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-49-320.jpg)
![30 Introduction and Preliminaries
4. Now let k = 1, . . . ,κ be arbitrary. Refer to Lemma 3.3.1 for the basic
properties of Lipschitz constants. Then, in view of the defining equality 3.3.2, the
function ηk has Lipschitz constant ε−1. Hence g+
k ≡ η0 ∨ · · · ∨ ηk has Lipschitz
constant ε−1. In particular,
y∈A gy ≡ g+
κ has Lipschitz constant ε−1. Moreover,
for each k = 1, . . . ,κ, the function
y∈A;yx(k)
gy ≡
k−1
i=1
gx(i) = g+
k
has Lipschitz constant ε−1, whence gx(k) ≡ g+
k − g+
k−1 has Lipschitz constant
2ε−1. Summing up, for each x ∈ A, the functions
y∈A gy,
y∈A;yx gy, and gx
have Lipschitz constant c ≡ 2ε−1. Assertion 4 and the proposition are proved.
Recall that if f ∈ C(S,d), then supx∈S |f (x)| exists and is denoted by f .
Definition 3.3.4. Partition of unity of a locally compact metric space. Let
(S,d) be a locally compact metric space, with a reference point x◦ ∈ S. Let the
nondecreasing sequence ξ ≡ (An)n=1,2,... of enumerated finite subsets of (S,d)
be a binary approximation of (S,d) relative to x◦.
For each n ≥ 1, let {gn,x : x ∈ An} be the 2−n-partition of unity of (S,d)
determined by the metrically discrete and finite subset An, as in Definition 3.3.2.
Then the sequence
π ≡ ({gn,x : x ∈ An})n=1,2,...
is called a partition of unity of (S,d) determined by the binary approximation ξ.
Proposition 3.3.5. Properties of partition of unity. Let ξ ≡ (An)n=1,2,... be
a binary approximation of the locally compact metric space (S,d) relative to a
reference point x◦. Let π ≡ ({gn,x : x ∈ An})n=1,2,... be the partition of unity
determined by ξ. Let n ≥ 1 be arbitrary. Then the following conditions hold:
1. gn,x ∈ C(S,d) has values in [0,1] and has support (d(·,x) ≤ 2−n+1), for
each x ∈ An.
2.
x∈A(n) gn,x ≤ 1 on S.
3.
x∈A(n) gn,x = 1 on
x∈A(n)(d(·,x) ≤ 2−n).
4. For each x ∈ An, the functions gn,x,
y∈A(n);yx gn,y, and
y∈A(n) gn,y
have Lipschitz constant 2n+1. Here y x means y = xi and x = xj for some
i,j ∈ {1, . . . ,|An|} with i j.
5. For each x ∈ An,
gn,x =
y∈A(n+1)
gn,xgn+1,y (3.3.5)
on S.
Proof. Assertions 1–4 are restatements of their counterparts in Proposition 3.3.3
for the case ε ≡ 2−n.](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-50-320.jpg)
![34 Introduction and Preliminaries
f −
x∈A(k)
f (x)gx
≤ ε, (3.3.11)
where the function g ≡
x∈A(k) f (x)gx has Lipschitz constant
x∈A(k)
|f (x)|2k+1
≤ 2k+1
|Ak|,
according to Assertion 2 of Lemma 3.3.1. Assertion 4 of the present proposition
is proved.
3.4 One-Point Compactification
The countable power of a locally compact metric space (S,d) is not necessarily
locally compact, while the countable power of a compact metric space remains
compact. For that reason, we will often find it convenient to embed a locally
compact metric space into a compact metric space such that while the metric is
not preserved, the continuous functions are. This embedding is made precise in
the present section, by an application of partitions of unity.
The next definition is an embellishment of definition 6.6, proposition 6.7, and
theorem 6.8 of [Bishop and Bridges 1985].
Definition 3.4.1. One-point compactification. A one-point compactification of
a locally compact metric space (S,d) is a compact metric space (S,d) with an
element , called the point at infinity, such that the following five conditions hold:
1. d ≤ 1 and S ∪ {} is a dense subset of (S,d).
2. Let K be an arbitrary compact subset of (S,d). Then there exists c 0 such
that d(x,) ≥ c for each x ∈ K.
3. Let K be an arbitrary compact subset of (S,d). Let ε 0 be arbitrary. Then
there exists δK(ε) 0 such that for each y ∈ K and z ∈ S with d(y,z) δK(ε),
we have d(y,z) ε. In particular, the identity mapping ῑ : (S,d) → (S,d),
defined by ι(x) ≡ x for each x ∈ S, is uniformly continuous on each compact
subset K of (S,d).
4. The identity mapping ι : (S,d) → (S,d), defined by ι(x) ≡ x for each
x ∈ S, is uniformly continuous on (S,d). In other words, for each ε 0, there
exists δd(ε) 0 such that d(x,y) ε for each x,y ∈ S with d(x,y) δd(ε).
5. For each n ≥ 1, we have
(d(·,x◦) 2n+1
) ⊂ (d(·,) ≤ 2−n
).
Thus, as a point x ∈ S moves away from x◦ relative to d, it converges to the point
at infinity relative to d.
First we provide some convenient notations.
Definition 3.4.2. Restriction of a family of functions. Let A,A be arbitrary
sets and let B be an arbitrary subset of A. Recall that the restriction of a function](https://image.slidesharecdn.com/16676206-250601161538-b43fa4bc/85/Foundations-Of-Constructive-Probability-Theory-Yuenkwok-Chan-54-320.jpg)
Foundations Of Constructive Probability Theory Yuenkwok Chan
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- 7. Foundations of Constructive Probability Theory Using Bishop’s work on constructive analysis as a framework, this monograph gives a systematic, detailed, and general constructive theory of probability theory and stochastic processes. It is the first extended account of this theory: Almost all of the constructive existence and continuity theorems that permeate the book are original. It also contains results and methods hitherto unknown in the constructive and nonconstructive settings. The text features logic only in the common sense and, beyond a certain mathematical maturity, requires no prior training in either constructive mathematics or probability theory. It will thus be accessible and of interest to both probabilists interested in the foundations of their specialty and constructive mathematicians who wish to see Bishop’s theory applied to a particular field. Yuen-Kwok Chan completed a PhD in constructive mathematics with Errett Bishop before leaving academia for a career in private industry. He is now an independent researcher in probability and its applications.
- 8. ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing, visit www.cambridge.org/mathematics 129 J. Berstel, D. Perrin, and C. Reutenauer Codes and Automata 130 T. G. Faticoni Modules over Endomorphism Rings 131 H. Morimoto Stochastic Control and Mathematical Modeling 132 G. Schmidt Relational Mathematics 133 P. Kornerup and D. W. Matula Finite Precision Number Systems and Arithmetic 134 Y. Crama and P. L. Hammer (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering 135 V. Berthé and M. Rigo (eds.) Combinatorics, Automata, and Number Theory 136 A. Kristály, V. D. Rădulescu, and C. Varga Variational Principles in Mathematical Physics, Geometry, and Economics 137 J. Berstel and C. Reutenauer Noncommutative Rational Series with Applications 138 B. Courcelle and J. Engelfriet Graph Structure and Monadic Second-Order Logic 139 M. Fiedler Matrices and Graphs in Geometry 140 N. Vakil Real Analysis through Modern Infinitesimals 141 R. B. Paris Hadamard Expansions and Hyperasymptotic Evaluation 142 Y. Crama and P. L. Hammer Boolean Functions 143 A. Arapostathis, V. S. Borkar, and M. K. Ghosh Ergodic Control of Diffusion Processes 144 N. Caspard, B. Leclerc, and B. Monjardet Finite Ordered Sets 145 D. Z. Arov and H. Dym Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations 146 G. Dassios Ellipsoidal Harmonics 147 L. W. Beineke and R. J. Wilson (eds.) with O. R. Oellermann Topics in Structural Graph Theory 148 L. Berlyand, A. G. Kolpakov, and A. Novikov Introduction to the Network Approximation Method for Materials Modeling 149 M. Baake and U. Grimm Aperiodic Order I: A Mathematical Invitation 150 J. Borwein et al. Lattice Sums Then and Now 151 R. Schneider Convex Bodies: The Brunn-Minkowski Theory (Second Edition) 152 G. Da Prato and J. Zabczyk Stochastic Equations in Infinite Dimensions (Second Edition) 153 D. Hofmann, G. J. Seal, and W. Tholen (eds.) Monoidal Topology 154 M. Cabrera García and Á. Rodríguez Palacios Non-Associative Normed Algebras I: The Vidav-Palmer and Gelfand-Naimark Theorems 155 C. F. Dunkl and Y. Xu Orthogonal Polynomials of Several Variables (Second Edition) 156 L. W. Beineke and R. J. Wilson (eds.) with B. Toft Topics in Chromatic Graph Theory 157 T. Mora Solving Polynomial Equation Systems III: Algebraic Solving 158 T. Mora Solving Polynomial Equation Systems IV: Buchberger Theory and Beyond 159 V. Berthé and M. Rigo (eds.) Combinatorics, Words, and Symbolic Dynamics 160 B. Rubin Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis 161 M. Ghergu and S. D. Taliaferro Isolated Singularities in Partial Differential Inequalities 162 G. Molica Bisci, V. D. Radulescu, and R. Servadei Variational Methods for Nonlocal Fractional Problems 163 S. Wagon The Banach-Tarski Paradox (Second Edition) 164 K. Broughan Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents 165 K. Broughan Equivalents of the Riemann Hypothesis II: Analytic Equivalents 166 M. Baake and U. Grimm (eds.) Aperiodic Order II: Crystallography and Almost Periodicity 167 M. Cabrera García and Á. Rodríguez Palacios Non-Associative Normed Algebras II: Representation Theory and the Zel’manov Approach 168 A. Yu. Khrennikov, S. V. Kozyrev, and W. A. Zúñiga-Galindo Ultrametric Pseudodifferential Equations and Applications 169 S. R. Finch Mathematical Constants II 170 J. Krajíček Proof Complexity 171 D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen Quasi-Hopf Algebras 172 P. McMullen Geometric Regular Polytopes 173 M. Aguiar and S. Mahajan Bimonoids for Hyperplane Arrangements 174 M. Barski and J. Zabczyk Mathematics of the Bond Market: A Lévy Processes Approach 175 T. R. Bielecki, J. Jakubowski, and M. Niewȩgłowski Fundamentals of the Theory of Structured Dependence between Stochastic Processes 176 A. A. Borovkov Asymptotic Analysis of Random Walks: Light-Tailed Distributions 177 Y.-K. Chan Foundations of Constructive Probability Theory
- 9. Foundations of Constructive Probability Theory YUEN-KWOK CHAN Citigroup
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108835435 DOI: 10.1017/9781108884013 © Yuen-Kwok Chan 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Chan, Yuen-Kwok, author. Title: Foundations of constructive probability theory / Yuen-Kwok Chan. Description: Cambridge, UK ; New York, NY : Cambridge University Press, 2021. | Series: Encyclopedia of mathematics and its applications | Includes bibliographical references and index. Identifiers: LCCN 2020046705 (print) | LCCN 2020046706 (ebook) | ISBN 9781108835435 (hardback) | ISBN 9781108884013 (epub) Subjects: LCSH: Probabilities. | Stochastic processes. | Constructive mathematics. Classification: LCC QA273 .C483 2021 (print) | LCC QA273 (ebook) | DDC 519.2–dc23 LC record available at https://lccn.loc.gov/2020046705 LC ebook record available at https://lccn.loc.gov/2020046706 ISBN 978-1-108-83543-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Dedicated to the memory of my father, Tak-Sun Chan
- 13. Contents Acknowledgments page x Nomenclature xi PART I INTRODUCTION AND PRELIMINARIES 1 1 Introduction 3 2 Preliminaries 6 2.1 Natural Numbers 6 2.2 Calculation and Theorem 6 2.3 Proof by Contradiction 7 2.4 Recognizing Nonconstructive Theorems 7 2.5 Prior Knowledge 7 2.6 Notations and Conventions 8 3 Partition of Unity 17 3.1 Abundance of Compact Subsets 17 3.2 Binary Approximation 19 3.3 Partition of Unity 27 3.4 One-Point Compactification 34 PART II PROBABILITY THEORY 43 4 Integration and Measure 45 4.1 Riemann–Stieljes Integral 45 4.2 Integration on a Locally Compact Metric Space 47 4.3 Integration Space: The Daniell Integral 51 4.4 Complete Extension of Integration 54 4.5 Integrable Set 65 4.6 Abundance of Integrable Sets 73 4.7 Uniform Integrability 88 vii
- 14. viii Contents 4.8 Measurable Function and Measurable Set 94 4.9 Convergence of Measurable Functions 109 4.10 Product Integration and Fubini’s Theorem 119 5 Probability Space 138 5.1 Random Variable 138 5.2 Probability Distribution on Metric Space 151 5.3 Weak Convergence of Distributions 155 5.4 Probability Density Function and Distribution Function 164 5.5 Skorokhod Representation 169 5.6 Independence and Conditional Expectation 182 5.7 Normal Distribution 189 5.8 Characteristic Function 201 5.9 Central Limit Theorem 220 PART III STOCHASTIC PROCESS 225 6 Random Field and Stochastic Process 227 6.1 Random Field and Finite Joint Distributions 227 6.2 Consistent Family of f.j.d.’s 231 6.3 Daniell–Kolmogorov Extension 241 6.4 Daniell–Kolmogorov–Skorokhod Extension 262 7 Measurable Random Field 275 7.1 Measurable r.f. That Is Continuous in Probability 275 7.2 Measurable Gaussian Random Field 290 8 Martingale 299 8.1 Filtration and Stopping Time 299 8.2 Martingale 306 8.3 Convexity and Martingale Convergence 314 8.4 Strong Law of Large Numbers 323 9 a.u. Continuous Process 331 9.1 Extension from Dyadic Rational Parameters to Real Parameters 333 9.2 C-Regular Family of f.j.d.’s and C-Regular Process 337 9.3 a.u. Hoelder Process 346 9.4 Brownian Motion 353 9.5 a.u. Continuous Gaussian Process 362 10 a.u. Càdlàg Process 374 10.1 Càdlàg Function 375 10.2 Skorokhod Space D[0,1] of Càdlàg Functions 384 10.3 a.u. Càdlàg Process 405 10.4 D-Regular Family of f.j.d.’s and D-Regular Process 409
- 15. Contents ix 10.5 Right-Limit Extension of D-Regular Process Is a.u. Càdlàg 413 10.6 Continuity of the Right-Limit Extension 433 10.7 Strong Right Continuity in Probability 443 10.8 Sufficient Condition for an a.u. Càdlàg Martingale 458 10.9 Sufficient Condition for Right-Hoelder Process 464 10.10 a.u. Càdlàg Process on [0,∞) 478 10.11 First Exit Time for a.u. Càdlàg Process 488 11 Markov Process 493 11.1 Markov Process and Strong Markov Process 494 11.2 Transition Distribution 495 11.3 Markov Semigroup 500 11.4 Markov Transition f.j.d.’s 502 11.5 Construction of a Markov Process from a Semigroup 510 11.6 Continuity of Construction 524 11.7 Feller Semigroup and Feller Process 536 11.8 Feller Process Is Strongly Markov 548 11.9 Abundance of First Exit Times 561 11.10 First Exit Time for Brownian Motion 568 APPENDICES 575 Appendix A Change of Integration Variables 577 Appendix B Taylor’s Theorem 605 References 606 Index 609
- 16. Acknowledgments Yuen-Kowk Chan is retired from Citigroup’s Mortgage Analytics unit. All opinions expressed by the author are his own. The author is grateful to the late Professor E. Bishop for teaching him constructive mathematics, to the late Professors R. Getoor and R. Blumenthal for teaching him probability and for mentoring him, and to the late Professors R. Pyke and W. Birnbaum and the other statisticians in the Mathematics Department of the University of Washington, circa 1970s, for their moral support. The author is also thankful to the constructivists in the Mathematics Department of New Mexico State University, circa 1975, for hosting a sabbatical visit and for valuable discussions, especially to Professors F. Richman, D. Bridges, M. Mandelkern, W. Julian, and the late Professor R. Mines. Professors Melody Chan and Fritz Scholz provided incisive and valuable critiques of the introduction chapter of an early draft of this book. Professor Douglas Bridges gave many thoughtful comments of the draft. The author also wishes to thank Ms Jill Hobbs for her meticulous copyediting, and Ms Niranjana Harikrishnan for her aesthetically pleasing typography. x
- 17. Nomenclature ≡........................................... by definition equal to, 8 R........................................... set of real numbers, 8 decld...................................... Euclidean metric, 8 a ∨ b..................................... max(a,b),8 a ∧ b..................................... min(a,b),8 a+......................................... max(a,0),8 a−......................................... min(a,0), 8 A ∪ B.................................... union of sets A and B, 8 A ∩ B,AB.............................intersection of sets A and B, 8 [a]1........................................an integer [a]1 ∈ (a,a + 2) for given a ∈ R, 9 X|A....................................... restriction of function X on a set to a subset A, 9 X ◦ X,X (X)....................... composite of functions X and X, 10 (X ≤ a)................................. {ω ∈ domain(X) : X(ω) ≤ a}, 11 X(·,ω ).... ............................ function of first variable, given value of second variable for a function X of two variables, 12 T ∗(Y) ≡ T (·,Y)................... dual function of Y relative to a certain mapping T , 12 (S,d) .....................................metric space, with metric d on set S, 12 x y .................................... d(x,y) 0, where x,y are in some metric space, 13 Jc ..........................................metric complement of subset J in a metric space, 12 .......................................... direct product of functions or sets, 14 Cu(S,d),Cu(S).....................space of uniformly continuous real-valued functions on metric space (S,d), 14 Cub(S,d),Cub(S)................. subspace of Cu(S,d) whose members are bounded, 14 C0(S,d), C0(S).....................subspace of Cu(S,d) whose members vanish at infinity, 15 C(S,d),C(S).........................subspace of Cu(S,d) whose members have bounded supports, 15 d............................................ 1 ∧ d, 15 O,o........................................bounds for real-valued function, 16 ............................................ mark for end of proof or end of definition, 16 ξ............................................ binary approximation of a metric space, 20 ξ ........................................ modulus of local compactness corresponding to ξ, 20 (S,d)..................................... one-point compactification of (S,d), 34 ........................................... point at infinity, 34 F|B....................................... {f |B : f ∈ F}, 35 +∞ −∞ X(x)dF(x)..................Riemann–Stieljes integral, 46 1A..........................................indicator of measurable set A, 67 Ac..........................................measure-theoretic complement of measurable set A, 67 ♦........................................... ordering between certain real numbers and functions, 73 xi
- 18. xii Nomenclature (G,λ).....................................profile system, 73 (a,b) α............................. the interval (a,b) is bounded in profile by α, 74 (,L,E)................................probability space, 138 E(dω)X(ω)....................... E(X), 139 ρP rob.................................... the probability metric on r.v.’s, 145 L(G)..................................... probability subspace generated by the family G of r.v.’s, 150 J(S,d)...................................set of distributions on complete metric space (S,d), 151 ⇒.......................................... weak convergence of distributions, or convergence in distributions of r.r.v.’s, 155 ρDist,ξ .................................. metric on distributions on a locally compact metric space relative to binary approximation ξ, 156 FX......................................... P.D.F. induced on R by an r.r.v. X, 165 Sk,ξ .................................... Skorokhod representation of distributions on (S,d), determined by ξ, 170 L|L ....................................... subspace of conditionally integrable r.r.v.’s given the subspace L , 184 L|G........................................subspace of conditionally integrable r.r.v.’s, given L(G), 184 EA.........................................conditional expectation given an event A with positive probability, 184 ϕμ,σ ...................................... multivariate normal p.d.f., 192 μ,σ ..................................... multivariate normal distribution, 193 ϕ0,I ....................................... multivariate standard normal p.d.f., 193 0,I ...................................... multivariate standard normal distribution, 193 ........................................... tail of univariate standard normal distribution, 193 ψX.........................................characteristic function of r.v. X with values in Rn, 204 ψJ ......................................... characteristic function of distribution J on Rn, 204 g............................................ Fourier transform of complex-valued function g on Rn, 204 f g......................................Convolution of complex-valued functions f and g on Rn, 204 ρchar ..................................... metric on characteristic functions on Rn, 210 R(Q × ,S)......................... set of r.f.’s with parameter set Q, state space (S,d), and sample space (,L,E), 227 X|K...................................... restriction of X ∈ R(Q × ,S) to parameter subset K ⊂ Q, 227 δCp,K ....................................modulus of continuity in probability of X|K, 228 δcau,K ................................... modulus of continuity a.u. of X|K, 228 δauc,K ................................... modulus of a.u. continuity of X|K, 228 F(Q,S)................................. set of consistent families of f.j.d.’s with parameter set Q and state space S, 232 ρMarg,ξ,Q............................ marginal metric for the set F(Q,S) relative to the binary approximation ξ, 237 FCp(Q,S)............................. subset of F(Q,S) whose members are continuous in probability, 238 ρCp,ξ,Q,Q(∞)...................... metric on FCp(Q,S) relative to dense subset Q∞of parameter metric space Q, 240 ρP rob,Q................................ probability metric on R(Q × ,S), 265 ρSup,P rob............................. metric on FCp(Q,S), 277 RCp(Q × ,S).....................subset of R(Q×,S) whose members are continuous in probability, 228 RMeas(Q × ,S).................subset of R(Q × ,S) whose members are measurable, 276 RMeas,Cp(Q × ,S)........... RMeas(Q × ,S) ∩ RCp(Q × ,S), 276 L............................................filtration in probability space (,L,E), 300 LX........................................ natural filtration of a process X, 300 L+........................................ right-limit extension of filtration L, 301 L(τ)....................................... probability subspace of observables at stopping time τ relative to filtra- tion L, 302 λ............................................ the special convex function on R, 316 Qm,Qm, Qm,Q∞,Q∞........certain subsets of dyadic rationals in [0,∞), 332 ( C[0,1],ρ C[0,1])................... metric space of a.u. continuous processes on [0,1], 334
- 19. Nomenclature xiii Lim.........................................................extension by limit of a process with parameter set Q∞ to parameter set [0,1], 335 D[0,1]....................................................... set of all a.u. càdlàg processes on [0,1], 406 δaucl.......................................................... modulus of a.u. càdlàg, 406 Dδ(aucl),δ(cp)[0,1]....................................subset of D[0,1] whose members have moduli δCp, and δaucl, 406 ρ D[0,1]...................................................... metric on D[0,1], 409 ( RDreg(Q∞ × ,S), ρP rob,Q(∞))......... metric space of D-regular processes, 410 rLim....................................................... extension by right-limit of a process with parameter set Q∞ to parameter set [0,1], 418 βauB......................................................... modulus of a.u. boundedness, 445 δSRCp....................................................... modulus of strong right continuity in probability, 445 τf,a,N (X)................................................ certain first exit times by the process X, 488 T................................................................ a Markov semigroup, 500 δT.............................................................. a modulus of strong continuity of T, 500 αT............................................................. a modulus of smoothness of T, 500 F∗,T r(1),···,r(m)............................................. a finite joint transition distribution generated by T, 502 (T ,ρT )................................................... metric space of Markov semigroups. 525 V............................................................... a Feller semigroup, 538 δV.............................................................. a modulus of strong continuity of V, 538 αV............................................................. a modulus of smoothness of V, 538 κV..............................................................a modulus of nonexplosion of V, 538 F∗,V r(1),···,r(m)..............................................a finite joint transition distribution generated by V, 539 ((S,d),(,L,E),{Ux,V : x ∈ S})............ Feller process, 543 ((Rm,dm),(,L,E),{Bx : x ∈ Rm})...... Brownian motion as a Feller process, 568
- 21. Part I Introduction and Preliminaries
- 23. 1 Introduction The present work on probability theory is an outgrowth of the constructive analysis in [Bishop 1967] and [Bishop and Bridges 1985]. Perhaps the simplest explanation of constructive mathematics is by way of focusing on the following two commonly used theorems. The first, the principle of finite search, states that, given a finite sequence of 0-or-1 integers, either all members of the sequence are equal to 0, or there exists a member that is equal to 1. We use this theorem without hesitation because, given the finite sequence, a finite search would determine the result. The second theorem, which we may call the principle of infinite search, states that, given an infinite sequence of 0-or-1 integers, either all members of the sequence are equal to 0, or there exists a member that is equal to 1. The name “infinite search” is perhaps unfair, but it brings into sharp focus the point that the computational meaning of this theorem is not clear. The theorem is tantamount to an infinite loop in computer programming without any assurance of termination. Most mathematicians acknowledge the important distinction between the two theorems but regard the principle of infinite search as an expedient tool to prove theorems, with the belief that theorems so proved can then be specialized to constructive theorems, when necessary. Contrary to this belief, many clas- sical theorems proved directly or indirectly via the principle of infinite search are actually equivalent to the latter: as such, they do not have a constructive proof. Oftentimes, not even the numerical meaning of the theorems in question is clear. We believe that, for the constructive formulations and proofs of even the most abstract theorems, the easiest way is to employ a disciplined and systematic approach, by using only finite searches and by quantifying mathematical objects and theorems at each and every step, with natural numbers as a starting point. The references cited earlier show that this approach is not only possible but also fruitful. It should be emphasized that we do not claim that theorems whose proofs require the principle of infinite search are untrue or incorrect. They are certainly correct and consistent derivations from commonly accepted axioms. There is 3
- 24. 4 Introduction and Preliminaries indeed no reason why we cannot discuss such classical theorems alongside their constructive counterparts. The term “nonconstructive mathematics” is not meant to be pejorative. We will use, in its place, the more positive term “classical mathematics.” Moreover, it is a myth that constructivists use a different system of logic. The only logic we use is commonsense logic; no formal language is needed. The present author considers himself a mathematician who is neither interested in nor equipped to comment on the formalization of mathematics, whether classical or constructive. Since a constructively valid argument is also correct from the classical view- point, a reader of the classical persuasion should have no difficulties understanding our proofs. Proofs using only finite searches are surely agreeable to any reader who is accustomed to infinite searches. Indeed, the author would consider the present book a success if the reader, but for this introduction and occasional remarks in the text, finishes reading without realizing that this is a constructive treatment. At the same time, we hope that a reader of the classical persuasion might consider the more disciplined approach of constructive mathematics for his or her own research an invitation to a challenge. Cheerfully, we hasten to add that we do not think that finite computations in constructive mathematics are the end. We would prefer a finite computation with n steps to one with n! steps. We would be happy to see a systematic and general development of mathematics that is not only constructive but also computationally efficient. That admirable task will, however, be left to abler hands. Probability theory, which is rooted in applications, can naturally be expected to be constructive. Indeed, the crowning achievements of probability theory – the laws of large numbers, the central limit theorems, the analysis of Brownian motion processes and their stochastic integrals, and the analysis of Levy processes, to name just a few – are exemplars of constructive mathematics. Kolmogorov, the grandfather of modern probability theory, actually took an interest in the formalization of general constructive mathematics. Nevertheless, many a theorem in modern probability actually implies the principle of infinite search. The present work attempts a systematic constructive development. Each existence theorem will be a construction. The input data, the construction procedure, and the output objects are the essence and integral parts of the theorem. Incidentally, by inspecting each step in the procedure, we can routinely observe how the output varies with the input. Thus a continuity theorem in epsilon–delta terms routinely follows an existence theorem. For example, we will construct a Markov process from a given semigroup and prove that the resulting Markov process varies continuously with the semigroup. The reader familiar with the probability literature will notice that our con- structions resemble Kolmogorov’s construction of the Brownian motion process, which is replete with Borel–Cantelli estimates and rates of convergence. This is in contrast to popular proofs of existence via Prokhorov’s theorem. The reader can
- 25. Introduction 5 regard Part III of this book, Chapters 6–11, the part on stochastic processes, as an extension of Kolmogorov’s constructive methods to stochastic processes: Daniell– Kolmogorov–Skorokhod construction of random fields, measurable random fields, a.u. continuous processes, a.u. càdlàg processes, martingales, strong Markov processes, and Feller processes, all with locally compact state spaces. Such a systematic, constructive, and general treatment of stochastic processes, we believe, has not previously been attempted. The purpose of this book is twofold. First, a student with a general mathematics background at the first-year graduate-school level can use it as an introduction to probability or to constructive mathematics. Second, an expert in probability can use it as a reference for further constructive development in his or her own research specialties. Part II of this book, Chapters 3–5, is a repackaging and expansion of the mea- sure theory in [Bishop and Bridges 1985]. It enables us to have a self-contained probability theory in terms familiar to probabilists. For expositions of constructive mathematics, see the first chapters of the last cited reference. See also [Richman 1982] and [Stolzenberg 1970]. We give a synopsis in Chapter 2, along with basic notations and terminologies for later reference.
- 26. 2 Preliminaries 2.1 Natural Numbers We start with the natural numbers as known in elementary schools. All mathemat- ical objects are constructed from natural numbers, and every theorem is ultimately a calculation on the natural numbers. From natural numbers are constructed the integers and the rational numbers, along with the arithmetical operations, in the manner taught in elementary schools. We claim to have a natural number only when we have provided a finite method to calculate it, i.e., to find its decimal representation. This is the fundamental difference from classical mathematics, which requires no such finite method; an infinite procedure in a proof is considered just as good in classical mathematics. The notion of a finite natural number is so simple and so immediate that no attempt is needed to define it in even simpler terms. A few examples would suffice as clarification: 1,2, and 3 are natural numbers. So are 99 and 999 ; the multiplication method will give, at least in principle, their decimal expansion in a finite number of steps. In contrast, the “truth value” of a particular mathematical statement is a natural number only if a finite method has been supplied that, when carried out, would prove or disprove the statement. 2.2 Calculation and Theorem An algorithm or a calculation means any finite, step-by-step procedure. A math- ematical object is defined when we specify the calculations that need to be done to produce this object. We say that we have proved a theorem if we have provided a step-by-step method that translates the calculations doable in the hypothesis to a calculation in the conclusion of the theorem. The statement of the theorem is merely a summary of the algorithm contained in the proof. Although we do not, for good reasons, write mathematical proofs in a computer language, the reader would do well to compare constructive mathematics to the development of a large computer software library, with successive objects and library functions being built from previous ones, each with a guarantee to finish in a finite number of steps. 6
- 27. Preliminaries 7 2.3 Proof by Contradiction There is a trivial form of proof by contradiction that is valid and useful in constructive mathematics. Suppose we have already proved that one of two given alternatives, A and B, must hold, meaning that we have given a finite method, that, when unfolded, gives either a proof for A or a proof for B. Suppose subse- quently we also prove that A is impossible. Then we can conclude that we have a proof of B; we need only exercise said finite method, and see that the resulting proof is for B. 2.4 Recognizing Nonconstructive Theorems Consider the simple theorem “if a is a real number, then a ≤ 0 or 0 a,” which may be called the principle of excluded middle for real numbers. We can see that this theorem implies the principle of infinite search by the following argument. Let (x)i=1,2,... be any given sequence of 0-or-1 integers. Define the real number a = ∞ i=1 xi2−i. If a ≤ 0, then all members of the given sequence are equal to 0; if 0 a, then some member is equal to 1. Thus the theorem implies the principle of infinite search, and therefore cannot have a constructive proof. Consequently, any theorem that implies this limited principle of excluded middle cannot have a constructive proof. This observation provides a quick test to recognize certain theorems as nonconstructive. Then it raises the interesting task of examining the theorem for constructivization of a part or the whole, or the task of finding a constructive substitute of the theorem that will serve all future purposes in its stead. For the aforementioned principle of excluded middle of real numbers, an adequate constructive substitute is the theorem “if a is a real number, then, for arbitrarily small ε 0, we have a ε or 0 a.” Heuristically, this is a recognition that a general real number a can be computed with arbitrarily small, but nonzero, error. 2.5 Prior Knowledge We assume that the reader of this book has familiarity with calculus, real analysis, and metric spaces, as well as some rudimentary knowledge of complex analysis. These materials are presented in the first chapters of [Bishop and Bridges 1985]. We will also quote results from typical undergraduate courses in calculus or linear algebra, with the minimal constructivization wherever needed. We assume also that the reader has had an introductory course in probability theory at the level of [Feller I 1971] or [Ross 2003]. The reader should have no difficulty in switching back and forth between constructive mathematics and classical mathematics, or at least no more than in switching back and forth between classical mathematics and computer programming. Indeed, the reader is urged to read, concurrently with this book if not before delving into it, the many classical texts in probability.
- 28. 8 Introduction and Preliminaries 2.6 Notations and Conventions If x,y are mathematical objects, we write x ≡ y to mean “x is defined as y,” “x, which is defined as y,” “x, which has been defined earlier as y,” or any other grammatical variation depending on the context. 2.6.1 Numbers Unless otherwise indicated, N,Q, and R will denote the set of integers, the set of rational numbers in the decimal or binary system, and the set of real numbers, respectively. We will also write {1,2, . . .} for the set of positive integers. The set R is equipped with the Euclidean metric d ≡ decld. Suppose a,b,ai ∈ R for i = m,m + 1, . . . for some m ∈ N. We will write limi→∞ ai for the limit of the sequence am,am+1, . . . if it exists, without explicitly referring to m. We will write a ∨ b,a ∧ b,a+, and a− for max(a,b), min(a,b),a ∨ 0, and a ∧ 0, respectively. The sum n i=m ai ≡ am + · · · + an is understood to be 0 if n m. The product n i=m ai ≡ am · · · an is understood to be 1 if n m. Suppose ai ≥ 0 for i = m, m + 1, . . . We write ∞ i=m ai ∞ if and only if ∞ i=m |ai| ∞, in which case ∞ i=m ai is taken to be limn→∞ n i=m ai. In other words, unless otherwise specified, convergence of a series of real numbers means absolute convergence. Regarding real numbers, we quote Lemma 2.18 from [Bishop and Bridges 1985], which will be used, extensively and without further comments, in the present book. Limited proof by contradiction of an inequality of real numbers. Let x,y be real numbers such that the assumption x y implies a contradiction. Then x ≤ y. This lemma remains valid if the relations and ≤ are replaced by and ≥, respectively. We note, however, that if the relations and ≤ are replaced by ≥ and , respectively, then the lemma would not have a constructive proof. Roughly speaking, the reason is that a constructive proof of x y implies the calculation of a positive ε 0 such that y − x ε, which is more than a proof of x ≤ y; the latter requires only a proof that x y is impossible and does not require the calculation of anything. The reader should ponder on the subtle but important difference. 2.6.2 Set, Operation, and Function Set. In general, a set is a collection of objects equipped with an equality relation. To define a set is to specify how to construct an element of the set, and how to prove that two elements are equal. A set is also called a family. A member ω in the collection is called an element of the latter, or, in symbols, ω ∈ . The usual set-theoretic notations are used. Let two subsets A and B of a set be given. We will write A∪B for the union, and A∩B or AB for the intersection. We write A ⊂ B if each member ω of A is a member of B. We write A ⊃ B for
- 29. Preliminaries 9 B ⊂ A. The set-theoretic complement of a subset A of the set is defined as the set {ω ∈ : ω ∈ A implies a contradiction}. We write ω A if ω ∈ A implies a contradiction. Nonempty set. A set is said to be nonempty if we can construct some element ω ∈ . Empty set. A set is said to be empty if it is impossible to construct an element ω ∈ . We will let φ denote an empty set. Operation. Suppose A,B are sets. A finite, step-by-step, method X that produces an element X(x) ∈ B given any x ∈ A is called an operation from A to B. The element X(x) need not be unique. Two different applications of the operation X with the same input element x can produce different outputs. An example of an operation is [·]1, which assigns to each a ∈ R an integer [a]1 ∈ (a,a + 2). This operation is a substitute of the classical operation [·] and will be used frequently in the present work. Function. Suppose , are sets. Suppose X is an operation that, for each ω in some nonempty subset A of , constructs a unique member X(ω) in . Then the operation X is called a function from to , or simply a function on . The subset A is called the domain of X. We then write X : → , and write domain(X) for the set A. Thus a function X is an operation that has the additional property that if ω1 = ω2 in domain(X), then X(ω1) = X(ω2) in . To specify a function X, we need to specify its domain as well as the operation that produces the image X(ω) from each given member ω of domain(X). Two functions X,Y are considered equal, X = Y in symbols, if domain(X) = domain(Y), and if X(ω) = Y(ω) for each ω ∈ domain(X). When emphasis is needed, this equality will be referred to as the set-theoretic equality, in contradistinction to almost everywhere equality, to be defined later. A function is also called a mapping. Domain of function on a set need not be the entire set. The nonempty domain(X) is not required to be the whole set . This will be convenient when we work with functions defined only almost everywhere, in a sense to be made precise later in the setting of a measure/integration space. Henceforth, we write X(ω) only with the implicit or explicit condition that ω ∈ domain(X). Miscellaneous set notations and function notations. Separately, we some- times use the expression ω → X(ω) for a function X whose domain is understood. For example, the expression ω → ω2 stands for the function X : R → R defined by X(ω) ≡ ω2 for each ω ∈ R. Let X : → be a function, and let A be a subset of such that A ∩ domain(X) is nonempty. Then the restriction X|A of X to A is defined as the function from A to with domain(X|A) ≡ A ∩ domain(X) and (X|A)(ω) for each ω ∈ domain(X|A). The set
- 30. 10 Introduction and Preliminaries B ≡ {ω ∈ : ω = X(ω) for some ω ∈ domain(X)} is called the range of the function X, and is denoted by range(X). A function X: A → B is called a surjection if range(X) = B; in that case, there exists an operation Y : B → A, not necessarily a function, such that X(Y(b) = b for each b ∈ B. The function X is called an injection if for each a,a ∈ domain(X) with X(a) = X(a ), we have a = a . It is called a bijection if domain(X) = A and if X is both a surjection and an injection. Let X : B → A be a surjection with domain(X) = B. Then the triple (A,B,X) is called an indexed set. In that case, we write Xb ≡ X(b) for each b ∈ B. We will, by abuse of notations, call A or {Xb : b ∈ B} an indexed set, and write A ≡ {Xb : b ∈ B}. We will call B the index set, and say that A is indexed by the members b of B. Finite set, enumerated set, countable set. A set A is said to be finite if there exists a bijection v : {1, . . . ,n} → A, for some n ≥ 1, in which case we write |A| ≡ n and call it the size of A. We will then call v an enumeration of the set A, and call the pair (A,v) an enumerated set. When the enumeration v is understood from the context, we will abuse notations and simply call the set A ≡ {v1, . . . ,vn} an enumerated set. A set A is said to be countable if there exists a surjection v : {1,2, . . .} → A. A set A is said to be countably infinite if there exists a bijection v : {1,2, . . .} → A. We will then call v an enumeration of the set A, and call the pair (A,v) an enumerated set. When the enumeration v is understood from the context, we will abuse notations and simply call the set A ≡ {v1,v2, . . .} an enumerated set. Composite function. Suppose X : → and X : → are such that the set A defined by A ≡ {ω ∈ domain(X) : X(ω) ∈ domain(X )} is nonempty. Then the composite function X ◦ X : → is defined to have domain(X ◦ X) = A and (X ◦ X)(ω) = X (X(ω)) for ω ∈ A. The alternative notation X (X) will also be used for X ◦ X. Sequence. Let be a set and let n ≥ 1 be an arbitrary integer. A function ω : {1, . . . ,n} → that assigns to each i ∈ {1, . . . ,n} an element ω(i) ≡ ωi ∈ is called a finite sequence of elements in . A function ω : {1,2, . . . ,} → that assigns to each i ∈ {1,2, . . .} an element ω(i) ≡ ωi ∈ is called an infinite sequence of elements in . We will then write ω ≡ (ω1, . . . ,ωn) ≡ or (ωi)i=1,...,n in the first case, and write (ω1,ω2, . . .) or (ωi)i=1,2,... in the second case, for the sequence ω. If, in addition, j is a sequence of integers in domain(ω), such that jk jh for each k h in domain(j), then the sequence ω◦j : domain(j) → is called a subsequence of ω. Throughout this book, we will write a subscripted symbol ab interchangeably with a(b) to lessen the burden on subscripts. Thus, ab(c) stands for abc . Similarly, ωjk ≡ ωj(k) ≡ ω(j(k)) for each k ∈ domain(j), and we write (ωj(1),ωj(2), . . .) or (ωj(k))k=1,2,..., or simply (ωj(k)), for the
- 31. Preliminaries 11 subsequence when the domain of j is clear. If (ω1, . . . ,ωn) is a sequence, we will write {ω1, . . . ,ωn} for the range of ω. Thus an element ω0 ∈ is in {ω1, . . . ,ωn} if and only if there exists i = 1, . . . ,n such that ω0 = ωi. Suppose (ωi)i=1,2,..., and (ωi)i=1,2,..., are two infinite sequences. We will write (ωi,ωi)i=1,2,... for the merged sequence (ω1,ω1,ω2,ω2, . . .). Similar notations are used for several sequences. Cartesian product of sequence of sets. Let (n)n=0,1,... be a sequence of nonempty sets. Consider any 0 ≤ n ≤ ∞, i.e., n is a nonnegative integer or the symbol ∞. We will let (n) denote the Cartesian product n j=0 j . Con- sider 0 ≤ k ∞ with k ≤ n. The coordinate function πk is the function with domain(πk) = (n) and πk(ω0,ω1, . . .) = ωk. If n = for each n ≥ 0, then we will write n for (n) for each n ≥ 0. Let X be a function on k and let Y be a function on (k). When confusion is unlikely, we will use the same symbol X also for the function X ◦πk on (n), which depends only on the kth coordinate. Likewise, we will use Y also for the function Y ◦ (π0, . . . ,πk) on (n), which depends only on the first k + 1 coordinates. Thus every function on k or (k) is identified with a function on (∞). Accordingly, sets of functions on k,(k) are regarded also as sets of functions on (n). Function of several functions. Let M be the family of all real-valued functions on , equipped with the set-theoretic equality for functions. Suppose X,Y ∈ M and suppose f is a function on R × R such that the set D ≡ {ω ∈ domain(X) ∩ domain(Y) : (X(ω),Y(ω)) ∈ domain(f )} is nonempty. Then f (X,Y) is defined as the function with domain(f (X,Y)) ≡ D and f (X,Y)(ω) ≡ f (X(ω),Y(ω)) for each ω ∈ D. The definition extends to a sequence of functions in the obvious manner. Examples are where f (x,y) ≡ x+y for each (x,y) ∈ R × R, or where f (x,y) ≡ xy for each (x,y) ∈ R × R. Convergent series of real-valued functions. Suppose (Xi)i=m,m+1,... is a sequence of real-valued functions on a set . Suppose the set D ≡ ω ∈ ∞ i=m domain(Xi) : ∞ i=m |Xi(ω)| ∞ is nonempty. Then ∞ i=m Xi is defined as the function with domain ∞ i=m Xi ≡ D and with value ∞ i=m Xi(ω) for each ω ∈ D. This function ∞ i=m Xi is then called a convergent series. Thus convergence for series means absolute convergence. Ordering of functions. Suppose X,Y ∈ M and A is a subset of , and suppose a ∈ R. We say X ≤ Y on A if (i) A ∩ domain(X) = A ∩ domain(Y) and (ii) X(ω) ≤ Y(ω) for each ω ∈ A ∩ domain(X). If X ≤ Y on , we will simply write X ≤ Y. Thus X ≤ Y implies domain(X) = domain(Y). We write X ≤ a if X(ω) ≤ a for each ω ∈ domain(X). We will write (X ≤ a) ≡ {ω ∈ domain(X) : X(ω) ≤ a}.
- 32. 12 Introduction and Preliminaries We make similar definitions when the relation ≤ is replaced by , ≥, , or =. We say X is nonnegative if X ≥ 0. Suppose a ∈ R. We will abuse notations and write a also for the constant function X with domain(X) = and with X(ω) = a for each ω ∈ domain(X). Regarding one of several variables as a parameter. Let X be a function on the product set × . Let ω ∈ be such that (ω ,ω ) ∈ domain(X) for some ω ∈ . Define the function X(ω ,·) on by domain(X(ω ,·)) ≡ {ω ∈ : (ω ,ω ) ∈ domain(X)}, and by X(ω ,·)(ω ) ≡ X(ω ,ω ) for each ω ∈ domain(X(ω ,·)). Similarly, let ω ∈ be such that (ω ,ω ) ∈ domain(X) for some ω ∈ . Define the function X(·,ω ) on by domain(X(·,ω )) ≡ {ω ∈ : (ω ,ω ) ∈ domain(X)}, and by X(·,ω )(ω ) ≡ X(ω ,ω ) for each ω ∈ domain(X(·,ω )). More generally, given a function X on the Cartesian product × × · · · × (n), for each (ω ,ω , . . . ,ω(n)) ∈ domain(X), we define similarly the functions X(·,ω ,ω , . . . ,ω(n)), X(ω , ·,ω , . . . ,ω(n)), . . .,X(ω ,ω , . . . ,ω(n−1),·) on the sets , , . . . ,(n), respectively. Let M ,M denote the families of all real-valued functions on two sets , , respectively, and let L be a subset of M ’. Suppose T : × L → R (2.6.1) is a real-valued function. We can define the function T ∗ : L → M with domain(T ∗ ) ≡ {Y ∈ L : domain(T (·,Y )) is nonempty} and by T ∗(Y) ≡ T (·,Y). When there is no risk of confusion, we write T also for the function T ∗, we write T Y for T (·,Y), and we write T : L → M interchangeably with the expression 2.6.1. Thus the duality T (·,Y)(ω ) ≡ T (ω ,Y) ≡ T (ω ,·)(Y). (2.6.2) 2.6.3 Metric Space The definitions and notations related to metric spaces in [Bishop and Bridges 1985], with few exceptions, are familiar to readers of classical texts. A summary of these definitions and notations follows. Metric complement. Let (S,d) be a metric space. If J is a subset of S, its metric complement is the set {x ∈ S : d(x,y) 0 for all y ∈ J}. Unless otherwise specified, Jc will denote the metric complement of J.
- 33. Preliminaries 13 Condition valid for all but countably many points in metric space. A con- dition is said to hold for all but countably many members of S if it holds for each member in the metric complement Jc of some countable subset J of S. Inequality in a metric space. We will say that two elements x,y ∈ S are unequal, and write x y, if d(x,y) 0. Metrically discrete subset of a metric space. We will call a subset A of S metrically discrete if, for each x,y ∈ A we have x = y or d(x,y) 0. Classically, each subset A of S is metrically discrete. Limit of a sequence of functions with values in a metric space. Let (fn)n=1,2,... be a sequence of functions from a set to S such that the set D ≡ ω ∈ ∞ i=1 domain(fi) : lim i→∞ fi(ω) exists in S is nonempty. Then limi→∞ fi is defined as the function with domain (limi→∞ fi) ≡ D and with value lim i→∞ fi (ω) ≡ lim i→∞ fi(ω) for each ω ∈ D. We emphasize that limi→∞ fi is well defined only if it can be shown that D is nonempty. Continuous function. A function f : S → S is said to be uniformly continuous on a subset A ⊂ domain(f ), relative to the metrics d,d on S,S respectively, if there exists an operation δ : (0,∞) → (0,∞) such that d (f (x),f (y)) ε for each x,y ∈ A with d(x,y) δ(ε), for each ε 0. When there is a need to be precise as to the metrics d,d , we will say that f : (S,d) → (S ,d ) is uniformly continuous on A. The operation δ is called a modulus of continuity of f on A. Lipschitz continuous function. If there exists a coefficient c ≥ 0 such that d (f (x),f (y)) ≤ cd(x,y) for all x,y ∈ A, then the function f is said to be Lipschitz continuous on A, and the constant c is then called a Lipschitz constant of f on A. In that case, we will say simply that f has Lipschitz constant c. Totally bounded metric space, compact metric space. A metric space (S,d) is said to be totally bounded if, for each ε 0, there exists a finite subset A ⊂ S such that for each x ∈ S there exists y ∈ A with d(x,y) ε. The subset A is then called an ε-approximation of S. A compact metric space K is defined as a complete and totally bounded metric space. Locally compact metric space. A subset A ⊂ S is said to be bounded if there exists x ∈ S and a 0 such that A ⊂ (d(·,x) ≤ a). A subset S ⊂ S is said to be locally compact if every bounded subset of S is contained in some compact subset. The metric space (S,d) is said to be locally compact if the subset S is locally compact. Continuous function on metric space. A function f : (S,d) → (S ,d ) is said to be continuous if domain(f ) = S and if it is uniformly continuous on each bounded subset K of S.
- 34. 14 Introduction and Preliminaries Product of a finite sequence of metric spaces. Suppose (Sn,dn)n=1,2,... is a sequence of metric spaces. For each integer n ≥ 1, define d(n) (x,y) ≡ n i=1 di (x,y) ≡ (d1 ⊗ · · · ⊗ dn)(x,y) ≡ n i=1 di(xi,yi) for each x,y ∈ n i=1 Si. Then (S(n) ,d(n) ) ≡ n i=1 (Si,di) ≡ n i=1 Si, n i=1 di is a metric space called the product metric space of S1, . . . ,Sn. Product of an infinite sequence of metric spaces. Define the infinite product metric ∞ i=1 di on ∞ i=1 Si by d(∞) (x,y) ≡ ∞ i=1 di (x,y) ≡ ∞ i=1 2−i (1 ∧ di(xi,yi)) for each x,y ∈ ∞ i=1 Si. Define the infinite product metric space (S(∞) ,d(∞) ) ≡ ∞ i=1 (Si,di) ≡ ∞ i=1 Si, ∞ i=1 di . Powers of a sequence of metric spaces. Suppose, in addition, (Sn,dn) is a copy of the same metric space (S,d) for each n ≥ 1. Then we simply write (Sn,dn) ≡ (S(n),d(n)) and (S∞,d∞) ≡ (S(∞),d(∞)). Thus, in this case, dn (x,y) ≡ n i=1 d(xi,yi) for each x = (x1, . . . ,xn),y = (y1, . . . ,yn) ∈ Sn, and d∞ (x,y) ≡ ∞ i=1 2−i (1 ∧ d(xi,yi)) for each x = (x1,x2, . . .),y = (y1,y2, . . .) ∈ S∞. If, in addition, (Sn,dn) is locally compact for each n ≥ 1, then the finite prod- uct space (S(n),d(n)) is locally compact for each n ≥ 1, while the infinite product space (S(∞),d(∞)) is complete but not necessarily locally compact. If (Sn,dn) is compact for each n ≥ 1, then both the finite and infinite product spaces are compact. Spaces of real-valued continuous functions. Suppose (S,d) is a metric space. We will write Cu(S,d), or simply Cu(S), for the space of real-valued functions functions on (S,d) with domain(f ) = S that are uniformly continuous on S. We will write Cub(S,d), or simply Cub(S), for the subspace of Cu(S) whose members are bounded. Let x◦ be an arbitrary, but fixed, reference point in (S,d). A uniformly continuous function f on (S,d) is then said to vanish at infinity if, for
- 35. Preliminaries 15 each ε 0, there exists a 0 such that |f | ≤ ε for each x ∈ S with d(x,x◦) a. Write C0(S,d), or simply C0(S), for the space of continuous functions on (S,d) that vanish at infinity. Space of real-valued continuous functions with bounded support. A real- valued function f on S is said to have a subset A ⊂ S as support if x ∈ domain(f ) and |f (x)| 0 together imply x ∈ A. Then we also say that f is supported by A, or that A supports f . We will write C(S,d), or simply C(S), for the subspace of Cu(S,d) whose members have bounded supports. In the case where (S,d) is locally compact, C(S) consists of continuous functions on (S,d) with compact supports. Summing up, C(S) ⊂ C0(S) ⊂ Cub(S) ⊂ Cu(S). Infimum and supremum. Suppose a subset A of R is nonempty. A number b ∈ R is called a lower bound of A, and A said to bounded from below, if b ≤ a for each a ∈ A. A lower bound b of A is called the greatest lower bound, or infimum, of A if b ≥ b for each lower bound b of A. In that case, we write inf A ≡ b. Similarly, a number b ∈ R is called an upper bound of A, and A said to be bounded from above, if b ≥ a for each a ∈ A. An upper bound b of A is called the least upper bound, or supremum, of A if b ≤ b for each upper bound b of A. In that case, we write sup A ≡ b. In contrast to classical mathematics, there is no constructive general proof for the existence of a infimum for an subset of R that is bounded from below. Existence needs to be proved before each usage for each special case, much as in the case of limits. In that regard, [Bishop and Bridges 1985] prove that if a nonempty subset A of R is totally bounded, then both inf A and sup A exist. Moreover, suppose f is a continuous real-valued function on a compact metric space (K,d). Then the last cited text proves that infK f ≡ inf{f (x) : x ∈ K} and supK f ≡ sup{f (x) : x ∈ K} exist. 2.6.4 Miscellaneous Notations for if, only if, etc. The symbols ⇒, ⇐, and ⇔ will in general stand for “only if,” “if,” and “if and only if,” respectively. An exception will be made where the symbol ⇒ is used for weak convergence, as defined later. The intended meaning will be clear from the context. Capping a metric at 1. If (S,d) is a metric space, we will write d ≡ 1 ∧ d. Abbreviation for subsets. We will often write “x,y, . . . ,z ∈ A” as an abbreviation for “{x,y, . . . ,z} ⊂ A.” Default notations for numbers. Unless otherwise indicated by the context, the symbols i,j,k,m,n,p will denote integers, the symbols a,b will denote real
- 36. 16 Introduction and Preliminaries numbers, and the symbols ε,δ will denote positive real numbers. For example, the statement “for each i ≥ 1” will mean “for each integer i ≥ 1.” Notations for convergence. Suppose (an)n=1,2,... is a sequence of real num- bers. Then an → a stands for limn→∞ an = a. We write an ↑ a if (an) is a nondecreasing sequence and an → a. Similarly, we write an ↓ a if (an) is a non- increasing sequence and an → a. More generally, suppose f is a function on some subset A ⊂ R. Then f (x) → a stands for limx→x0 f (x) = a, where x0 can stand for a real number or for one of the symbols ∞ or −∞. Big O and small o. Suppose f and g are functions with domains equal to some subset A ⊂ R. Let x0 ∈ A be arbitrary. If for some c 0, we have |f (x)| ≤ c|g(x)| for all x ∈ A in some neighborhood B of x0, then we write f (x) = O(g(x)). If for each ε 0, we have |f (x)| ≤ ε|g(x)| for each x ∈ A in some neighborhood B of x0, then we write f (x) = o(g(x)). Here, a subset B ⊂ R is a neighborhood of x0 if there exists an open interval (a,b) such that x0 ∈ (a,b). End-of-proof or end-of-definition marker. Finally, we sometimes use the symbol to mark the end of a proof or a definition.
- 37. 3 Partition of Unity In the previous chapter, we summarized the basic concepts and theorems about metric spaces from [Bishop and Bridges 1985]. Locally compact metric space was introduced. It is a simple and wide-ranging generalization of the real line. With few exceptions, the metric spaces used in the present book are locally compact. In the present chapter, we will define and construct binary approximations of a locally compact metric space (S,d), then define and construct a partition of unity relative to each binary approximation. Roughly speaking, a binary approximation is a digitization of (S,d), a generalization of the dyadic rationals that digitize the space R of real numbers. A partition of unity is then a sequence in C(S,d) that serves as a basis for the linear space C(S,d) of continuous functions on (S,d) with compact supports, in the sense that each f ∈ C(S,d) can be approximated by linear combinations of members in the partition of unity. A partition of unity provides a countable set of basis functions for the metriza- tion of probability distributions on the space (S,d). Because of that important role, we will study binary approximations and partitions of unity in detail in this chapter. 3.1 Abundance of Compact Subsets First we cite a theorem from [Bishop and Bridges 1985] that guarantees an abun- dance of compact subsets. Theorem 3.1.1. Abundance of compact sets. Let f : K → R be a continuous function on a compact metric space (K,d) with domain(f ) = K. Then, for all but countably many real numbers α infK f , the set (f ≤ α) ≡ {x ∈ K : f (x) ≤ α} is compact. Proof. See theorem (4.9) in chapter 4 of [Bishop and Bridges 1985]. Classically, the set (f ≤ α) is compact for each α ≥ infK f , without exception. Such a general theorem would, however, imply the principle of infinite search and is therefore nonconstructive. Theorem 3.1.1 is sufficient for all our purposes. 17
- 38. 18 Introduction and Preliminaries Definition 3.1.2. Convention for compact sets (f ≤ α). We hereby adopt the convention that if the compactness of the set (f ≤ α) is required in a discussion, compactness has been explicitly or implicitly verified, usually by proper prior selection of the constant α, enabled by an application of Theorem 3.1.1. The following simple corollary of Theorem 3.1.1 guarantees an abundance of compact neighborhoods of a compact set. Corollary 3.1.3. Abundance of compact neighborhoods. Let (S,d) be a locally compact metric space, and let K be a compact subset of S. Then the subset Kr ≡ (d(·,K) ≤ r) ≡ {x ∈ S : d(x,K) ≤ r} is compact for all but countably many r 0. Proof. 1. Let n ≥ 1 be arbitrary. Then An ≡ (d(·,K) ≤ n) is a bounded set. Since (S,d) is locally compact, there exists a compact set Kn such that An ⊂ Kn ⊂ S. The continuous function f on the compact metric space (Kn,d) defined by f ≡ d(·,K) has infimum 0. Hence, by Theorem 3.1.1, the set {x ∈ Kn : d(x,K) ≤ r} is compact for all but countably many r ∈ (0,∞). In other words, there exists a countable subset J of (0,∞) such that for each r in the metric complement Jc of J in (0,∞), the set {x ∈ Kn : d(x,K) ≤ r} is compact. 2. Now let r ∈ Jc be arbitrary. Take n ≥ 1 so large that r ∈ (0,n). Then Kr ⊂ An. Hence the set Kr = KrAn = KrKn = {x ∈ Kn : d(x,K) ≤ r} is compact according to Step 1. Since J is countable and r ∈ Jc is arbitrary, we see that Kr is compact for all but countably many r ∈ (0,∞). Separately, the next elementary metric-space lemma will be convenient. Lemma 3.1.4. If (S, d) is compact, then the subspace of C(S∞, d∞) consisting of members that depend on finitely many coordinates is dense in C(S∞, d∞). Suppose (S,d) is a compact metric space. Let x◦ be an arbitrary but fixed reference point in (S,d). Let n ≥ 1 be arbitrary. Define the projection mapping j∗ n : S∞ → S∞ by j∗ n (x1,x2, . . .) ≡ (x1,x2, . . . ,xn,x◦,x◦, . . .) for each (x1,x2, . . .) ∈ S∞. Then j∗ n ◦ j∗ m = j∗ n for each m ≥ n. Let L0,n ≡ {f ∈ C(S∞ ,d∞ ) : f = f ◦ j∗ n }. (3.1.1) Then L0,n ⊂ L0,n+1.
- 39. Partition of Unity 19 Let L0,∞ ≡ ∞ n=1 L0,n. Then the following conditions hold: 1. L0,n and L0,∞ are linear subspaces of C(S∞,d∞), and consist of func- tions that depend, respectively, on the first n coordinates and on finitely many coordinates. 2. The subspace L0,∞ is dense in C(S∞,d∞) relative to the supremum norm · . Specifically, let f ∈ C(S∞,d∞) be arbitrary, with a modulus of continuity δf . Then f ◦ j∗ n ∈ L0,n. Moreover, for each ε 0 we have f − f ◦ j∗ n ≤ ε if n − log2(δf (ε)). In particular, if f has Lipschitz constant c 0, then f − f ◦ j∗ n ≤ ε if n log2(cε−1). Proof. Let m ≥ n ≥ 1 and w ∈ S∞ be arbitrary. Then, for each (x1,x2, . . .) ∈ S∞, we have j∗ n (j∗ m(x1,x2, . . .)) = j∗ n (x1,x2, . . . ,xm,x◦,x◦, . . .) = (x1,x2, . . . ,xn,x◦,x◦, . . .) = j∗ n (x1,x2, . . .). Thus j∗ n ◦ j∗ m = j∗ n . 1. It is clear from the defining equality 3.1.1 that L0,n is a linear subspace of C(S∞,d∞). Let f ∈ L0,n be arbitrary. Then f = f ◦ j∗ n = f ◦ j∗ n ◦ j∗ m = f ◦ j∗ m. Hence f ∈ L0,m. Thus L0,n ⊂ L0,m. Consequently, L0,∞ ≡ ∞ p=1 L0,p is a union of a nondecreasing sequence of linear subspaces of C(S∞,d∞) and, therefore, is also a linear subspace of C(S∞,d∞). 2. Let f ∈ C(S∞,d∞) be arbitrary, with a modulus of continuity δf . Let ε 0 be arbitrary. Suppose n − log2(δf (ε)). Then 2−n δf (ε). Let (x1,x2, . . .) ∈ S∞ be arbitrary. Then d∞ ((x1,x2, . . .),j∗ n (x1,x2, . . .)) = d∞ ((x1,x2, . . .),(x1,x2, . . . ,xn,x◦,x◦, . . .)) ≡ n k=1 2−k d(xk,xk) + ∞ k=n+1 2−k d(xk,x◦) ≤ 0 + 2−n δf (ε), where d ≡ 1 ∧ d. Hence |f (x1,x2, . . .) − f ◦ j∗ n (x1,x2, . . .)| ε, where (x1,x2, . . .) ∈ S∞ is arbitrary. We conclude that f − f ◦ j∗ n ≤ ε, as alleged. 3.2 Binary Approximation Let (S,d) be an arbitrary locally compact metric space. Then S contains a countable dense subset. A binary approximation, defined presently, is a structured and well-quantified countable dense subset.
- 40. 20 Introduction and Preliminaries Recall that (i) |A| denotes the number of elements in an arbitrary finite set A; (ii) a subset A of S is said to be metrically discrete if for each y,z ∈ A, either y = z or d(y,z) 0; and (iii) a finite subset A of a subset K ⊂ S is called an ε-approximation of K if for each x ∈ K, there exists y ∈ A with that d(x,y) ≤ ε. Classically, each subset of (S,d) is metrically discrete. Condition (iii) can be written more succinctly as K ⊂ x∈A (d(·,x) ≤ ε). Definition 3.2.1. Binary approximation and modulus of local compactness. Let (S,d) be a locally compact metric space, with an arbitrary but fixed reference point x◦. Let A0 ≡ {x◦} ⊂ A1 ⊂ A2 ⊂ . . . be a sequence of metrically discrete and finite subsets of S. For each n ≥ 1, write κn ≡ |An|. Suppose (d(·,x◦) ≤ 2n ) ⊂ x∈A(n) (d(·,x) ≤ 2−n ) (3.2.1) and x∈A(n) (d(·,x) ≤ 2−n+1 ) ⊂ (d(·,x◦) ≤ 2n+1 ) (3.2.2) for each n ≥ 1. Then the sequence ξ ≡ (An)n=1,2,... of subsets is called a binary approximation for (S,d) relative to x◦, and the sequence of integers ξ ≡ (κn)n=1,2,... ≡ (|An|)n=1,2,... is called the modulus of local compactness of (S,d) corresponding to ξ. Thus a binary approximation is an expanding sequence of 2−n-approximation for (d(·,x◦) ≤ 2n) as n → ∞. The next proposition shows that the definition is not vacuous. First note that ∞ n=1 An is dense in (S,d) in view of relation 3.2.1. In the case where (S,d) is compact, for n ≥ 1 so large that S = (d(·,x◦) ≤ 2n), relation 3.2.1 says that we need at most κn points to make a 2−n-approximation of S.1 Lemma 3.2.2. Existence of metrically discrete ε-approximations. Let K be a compact subset of the locally compact metric space (S,d). Let A0 ≡ {x1, . . . ,xn} be an arbitrary metrically discrete finite subset of K. Let ε 0 be arbitrary. Then the following conditions hold: 1. There exists a metrically discrete finite subset A of K such that (i) A0 ⊂ A and (ii) A is an ε-approximation of K. 2. In particular, there exists a metrically discrete finite set A that is an ε-approximation of K. 1 Incidentally, the number log κn is a bound for Kolmogorov’s 2−n-entropy of the compact met- ric space (S,d), which represents the informational content in a 2−n-approximation of S. See [Lorentz 1966] for a definition of ε-entropy.
- 41. Partition of Unity 21 Proof. 1. By hypothesis, the set A0 ≡ {x1, . . . ,xn} is a metrically discrete finite subset of the compact set K. Let ε 0 be arbitrary. Take an arbitrary ε0 ∈ (0,ε). Take any ε0-approximation {y1, . . . ,ym} of K. Write α ≡ m−1(ε − ε0). 2. Trivially, the set A0 ∪ {y1, . . . ,ym} is an ε0-approximation of K. Moreover, A0 is metrically discrete. 3. Consider y1. Then either (i) there exists x ∈ A0 such that d(x,y1) α or (ii) for each x ∈ A0 we have d(x,y1) 0. In case (i), define A1 ≡ A0. In case (ii), define A1 ≡ A0∪{y1}. Then, in either case, A1 is a metrically discrete finite subset of K. Moreover, A0 ⊂ A1. By the assumption in Step 2, the set A0 ∪ {y1, . . . ,ym} is an ε0-approximation of K. Hence there exists z ∈ A0 ∪ {y1, . . . ,ym} such that d(z,y) ε0. There are three possibilities: (i ) z ∈ A0, (ii ) z ∈ {y2, . . . ,ym}, or (iii ) z = y1. 4. In cases (i ) and (ii ), we have, trivially, z ∈ A1 ∪ {y2, . . . ,ym} and d(z,y) ε0 + α. Let w ≡ z ∈ A0 ∪ {y1, . . . ,ym}. 5. Consider case (iii ), where z = y1. Then, according to Step 3, either (i) or (ii) holds. In case (ii), we have, again trivially, z = y1 ∈ A0 ∪ {y1} ≡ A1, and d(z,y) ε0. Consider case (i). Then A1 ≡ A0, and there exists x ∈ A0 such that d(x,y1) α. Consequently, x ∈ A1 and d(x,y) ≤ d(x,y1) + d(y1,y) = d(x,y1) + d(z,y) α + ε0. Let w ≡ x ∈ A0 ∪ {y1, . . . ,ym}. 6. Combining Steps 4 and 5, we see that, in any case, there exists w ∈ A1 ∪ {y2, . . . ,ym} such that d(w,y) ε0 + α. Since y ∈ K is arbitrary, we conclude that the set A1 ∪ {y2, . . . ,ym} is an (ε0 + α)-approximation of K, and that the set A1 is metrically discrete. Moreover, A0 ⊂ A1. 7. Repeat Steps 3–6 with A1 ∪ {y2, . . . ,ym} and ε0 + α in the roles of A0 ∪ {y1, . . . ,ym} and ε0, respectively. We obtain a metrically discrete subset A2 such that A2 ∪ {y3, . . . ,ym} is an (ε0 + 2α)-approximation of K. Moreover A0 ⊂ A2. 8. Recursively on y3, . . . ,ym, we obtain a metrically discrete subset Am that is an (ε0 + mα)-approximation of K. Moreover, A0 ⊂ Am. Since ε0 + mα ≡ ε0 + (ε − ε0) = ε,
- 42. 22 Introduction and Preliminaries it follows that Am is an ε-approximation of K. In conclusion, the set A ≡ Am has the desired properties in Assertion 1. 9. To prove Assertion 2, take an arbitrary x1 ∈ K and define A0 ≡ {x1}. Then, by Assertion 1, there exists a metrically discrete finite subset A of K such that (i) A0 ⊂ A and (ii) A is an ε-approximation of K. Thus the set A has the desired properties in Assertion 2. Proposition 3.2.3. Existence of binary approximations. Each locally compact metric space (S,d) has a binary approximation. Proof. Let x◦ ∈ S be an arbitrary but fixed reference point. Proceed inductively on n ≥ 0 to construct a metrically discrete and finite subset An of S to satisfy Conditions 3.2.1 and 3.2.2 in Definition 3.2.1. To start, let A0 ≡ {x◦}. Suppose the set An has been constructed for some n ≥ 0 such that if n ≥ 1, then (i) An is metrically discrete and finite and (ii) Conditions 3.2.1 and 3.2.2 in Definition 3.2.1 are satisfied. Proceed to construct An+1. To that end, write ε ≡ 2−n−2 and take any r ∈ [2n+1,2n+1 + ε) such that K ≡ (d(·,x◦) ≤ r) is compact. This is possible in view of Corollary 3.1.3. If n = 0, then An ≡ {x◦} ⊂ K trivially. If n ≥ 1, then, according to the induction hypothesis, the set An is metrically discrete, and by Condition 3.2.2 in Definition 3.2.1, we have An ⊂ x∈A(n) (d(·,x) ≤ 2−n+1 ) ⊂ (d(·,x◦) ≤ 2n+1 ) ⊂ K. Hence we can apply Lemma 3.2.2 to construct a 2−n−1-approximation An+1 of K, which is metrically discrete and finite, such that An ⊂ An+1. It follows that (d(·,x◦) ≤ 2n+1 ) ⊂ K ⊂ x∈A(n+1) (d(·,x) ≤ 2−n−1 ), proving Condition 3.2.1 in Definition 3.2.1 for n + 1. Now let y ∈ x∈A(n+1) (d(·,x) ≤ 2−n ) be arbitrary. Then d(y,x) ≤ 2−n for some x ∈ An+1 ⊂ K. Hence d(x,x◦) ≤ r 2n+1 + ε. Consequently, d(y,x◦) ≤ d(y,x) + d(x,x◦) ≤ 2−n + 2n+1 + ε ≡ 2−n + 2n+1 + 2−n−2 ≤ 2n+2 .
- 43. Partition of Unity 23 Thus we see that x∈A(n+1) (d(·,x) ≤ 2−n ) ⊂ (d(·,x◦) ≤ 2n+2 ), proving Condition 3.2.2 in Definition 3.2.1 for n+1. Induction is completed. The sequence ξ ≡ (An)n=1,2,... satisfies all the conditions in Definition 3.2.1 to be a binary approximation of (S,d). Definition 3.2.4. Finite product and power of binary approximations. Let n ≥ 1 be arbitrary. For each i = 1, . . . ,n, let (Si,di) be a locally compact metric space, with a reference point xi,◦ ∈ Si and with a binary approximation ξi ≡ (Ai,p)p=1,2,... relative to xi,◦. Let (S(n),d(n)) ≡ n i=1 Si, n i=1 di be the product metric space, with x(n) ◦ ≡ (x1,◦, . . . ,xn,◦) designated as the reference point in (S(n),d(n)). For each p ≥ 1, let A (n) p ≡ A1,p × · · · × An,p. The next lemma proves that (A (n) p )p=1,2,... is a binary approximation of (S(n),d(n)) relative to x (n) ◦ . We will call ξ(n) ≡ (A(n) p )p=1,2,... the product binary approximation of ξ1, . . . ,ξn, and write ξ(n) ≡ ξ1 ⊗ · · · ⊗ ξn. If (Si,di) = (S,d) for some locally compact metric space, with xi,◦ = x◦ and ξi = ξ for each i = 1, . . . ,n, then we will call ξ(n) the nth power of the binary approximation ξ, and write ξn ≡ ξ(n). Lemma 3.2.5. Finite product binary approximation is indeed a binary approximation. Use the assumptions and notations in Definition 3.2.4. Then the finite product binary approximation ξ(n) is indeed a binary approximation of (S(n),d(n)) relative to x (n) ◦ . Let ξi ≡ (κi,p)p=1,2,... ≡ (|Ai,p|)p=1,2,... be the modulus of local com- pactness of (Si,di) corresponding to ξi, for each i = 1, . . . ,n. Let ξ(n) be the modulus of local compactness of (S(n),d(n)) corresponding to ξ(n). Then ξ(n) = n i=1 κi,p p=1,2,... . In particular, if ξi ≡ ξ for each i = 1, . . . ,n for some binary approximation ξ of some locally compact metric space (S,d), then the finite power binary approx- imation is indeed a binary approximation of (Sn,dn) relative to xn ◦ . Moreover, the modulus of local compactness of (Sn,dn) corresponding to ξn is given by ξn = (κn p)p=1,2,.... Proof. Recall that A (n) p ≡ A1,p × · · · × An,p for each p ≥ 1. Hence A (n) 1 ⊂ A (n) 2 ⊂ · · · . 1. Let p ≥ 1 be arbitrary. Let x ≡ (x1, . . . ,xn),y ≡ (y1, . . . ,yn) ∈ A(n) p ≡ A1,p × · · · × An,p
- 44. 24 Introduction and Preliminaries be arbitrary. For each i = 1, . . . ,n, because (Ai,q)q=1,2,... is a binary approxima- tion of (Si,di), the set Ai,p is metrically discrete. Hence either (i) xi = yi for each i = 1, . . . ,n or (ii) di(xi,yi) 0 for some i = 1, . . . ,n. In case (i), we have x = y. In case (ii), we have d(n) (x,y) ≡ n j=1 dj (xj,yj ) ≥ di(xi,yi) 0. Thus the set A (n) p is metrically discrete. 2. Next note that (d(n) (·,x(n) ◦ ) ≤ 2p ) ≡ (y1, . . . ,yn) ∈ S(n) : n i=1 di(yi,xi,◦) ≤ 2p = n i=1 {(y1, . . . ,yn) ∈ S(n) : di(yi,xi,◦) ≤ 2p } ⊂ C ≡ n i=1 x(i)∈A(i,p) {(y1, . . . ,yn) ∈ S(n) : di(yi,xi) ≤ 2−p }, (3.2.3) where the last inclusion is due to Condition 3.2.1 applied to the binary approxi- mation (Ai,q)q=1,2,.... Basic Boolean operations then yield C = (x(1),...,x(n))∈A(1,p)×···×A(n,p) n i=1 {(y1, . . . ,yn) ∈ S(n) : di(yi,xi) ≤ 2−p } = (x(1),...,x(n))∈A(1,p)×···×A(n,p) (y1, . . . ,yn) ∈ S(n) : n i=1 di(yi,xi) ≤ 2−p ≡ (x(1),...,x(n))∈A(1,p)×···×A(n,p) × {(y1, . . . ,yn) ∈ S(n) : d(n) ((y1, . . . ,yn),(x1, . . . ,xn)) ≤ 2−p } ≡ (x(1),...,x(n))∈A(1,p)×···×A(n,p) (d(n) (·,(x1, . . . ,xn)) ≤ 2−p ) = x∈A (n) p (d(n) (·,x) ≤ 2−p ). (3.2.4) Combining with relation 3.2.3, this yields (d(n) (·,x(n) ◦ ) ≤ 2p ) ⊂ x∈A (n) p (d(n) (·,x) ≤ 2−p ), where p ≥ 1 is arbitrary. Thus Condition 3.2.1 has been verified for the sequence ξ(n) ≡ (A (n) p )p=1,2,....
- 45. Partition of Unity 25 3. In the other direction, we have, similarly, x∈A (n) p (d(n) (·,x) ≤ 2−p+1 ) ≡ x∈A (n) p n i=1 (di(·,x) ≤ 2−p+1 ) = n i=1 x(i)∈A(i,p) {(y1, . . . ,yn) ∈ S(n) : di(yi,xi) ≤ 2−p+1 } ⊂ n i=1 {(y1, . . . ,yn) ∈ S(n) : di(yi,xi,◦) ≤ 2p+1 } = (y1, . . . ,yn) ∈ S(n) : n i=1 di(yi,xi,◦) ≤ 2p+1 = (d(n) (·,x(n) ◦ ) ≤ 2p+1 ), where p ≥ 1 is arbitrary. This verifies Condition 3.2.2 for the sequence ξ(n) ≡ (A (n) p )p=1,2,.... Thus all the conditions in Definition 3.2.1 have been proved for the sequence ξ(n) to be a binary approximation of (S(n),d(n)) relative to x (n) ◦ . Moreover, ξ(n) ≡ (|A(n) q |)q=1,2,... = n i=1 |Ai,q| q=1,2,... ≡ n i=1 κi,q q=1,2,... . We next extend the construction of the powers of binary approximations to the infinite power (S∞,d∞) of a compact metric space (S,d). Recall that d ≡ 1 ∧ d. Definition 3.2.6. Countable power of binary approximation for a compact metric space. Suppose (S,d) is a compact metric space, with a reference point x◦ ∈ S and a binary approximation ξ ≡ (An)n=1,2,... relative to x◦. Let (S∞,d∞) be the countable power of the metric space (S,d), with x∞ ◦ ≡ (x◦,x◦, . . .) designated as the reference point in (S∞,d∞). For each n ≥ 1, define the subset Bn ≡ An+1 n+1 × {x∞ ◦ } = {(x1, . . . ,xn+1,x◦,x◦ · · · ) : xi ∈ An+1 for each i = 1, . . . ,n + 1}. The next lemma proves that ξ∞ ≡ (Bn)n=1,2,... is a binary approximation of (S∞,d∞) relative to x∞ ◦ . We will call ξ∞ the countable power of the binary approximation ξ. Lemma 3.2.7. Countable power of binary approximation of a compact metric space is indeed a binary approximation. Suppose (S,d) is a compact
- 46. 26 Introduction and Preliminaries metric space, with a reference point x◦ ∈ S and a binary approximation ξ ≡ (An)n=1,2,... relative to x◦. Without loss of generality, assume that d ≤ 1. Then the sequence ξ∞ ≡ (Bn)n=1,2,... in Definition 3.2.6 is indeed a binary approximation of (S∞,d∞) relative to x∞ ◦ . Let ξ ≡ (κn)n=1,2,... ≡ (|An|)n=1,2,... denote the modulus of local com- pactness of (S,d) corresponding to ξ. Then the modulus of local compactness of (S∞,d∞) corresponding to ξ∞ is given by ξ∞ = (κn+1 n+1 )n=1,2,.... Proof. Let n ≥ 1 be arbitrary. 1. Let x ≡ (x1, . . . ,xn+1,x◦,x◦, . . .),y ≡ (y1, . . . ,yn+1,x◦,x◦, . . .) ∈ Bn be arbitrary. Since An+1 is metrically discrete, we have either (i) xi = yi for each i = 1, . . . ,n + 1 or (ii) d(xi,yi) 0 for some i = 1, . . . ,n + 1. In case (i), we have x = y. In case (ii), we have d∞ (x,y) ≡ ∞ j=1 2−j d(xj,yj ) ≥ 2−i d(xi,yi) 0. Thus we see that Bn is metrically discrete. 2. Next, let y ≡ (y1,y2, . . .) ∈ S∞ be arbitrary. Let j = 1, . . . ,n + 1 be arbi- trary. Then yj ∈ (d(·,x◦) ≤ 2n+1 ) ⊂ z∈A(n+1) (d(·,z) ≤ 2−n−1 ), where the first containment relation is a trivial consequence of the hypothesis that d ≤ 1, and where the second is an application of Condition 3.2.1 of Definition 3.2.1 to the binary approximation ξ ≡ (An)n=1,2,.... Hence there exists some uj ∈ An+1 with d(yj,uj ) ≤ 2−n−1, where j = 1, . . . ,n + 1 is arbitrary. It follows that u ≡ (u1, . . . ,un+1,x◦,x◦, . . .) ∈ An+1 n+1 × {x∞ ◦ } ≡ Bn, and that d∞ (y,u) ≤ n+1 j=1 2−j d(yj,uj ) + ∞ j=n+2 2−j ≤ n+1 j=1 2−j 2−n−1 + 2−n−1 2−n−1 + 2−n−1 = 2−n , where y ∈ S∞ is arbitrary. We conclude that (d∞ (·,x∞ ◦ ) ≤ 2n ) = S∞ ⊂ u∈B(n) (d∞ (·,u) ≤ 2−n ).
- 47. Partition of Unity 27 where the equality is trivial because d∞ ≤ 1. Thus Condition 3.2.1 of Definition 3.2.1 is verified for the sequence (Bn)n=1,2,.... At the same time, again because because d∞ ≤ 1, we have trivially u∈B(n) (d∞ (·,u) ≤ 2−n+1 ) ⊂ S∞ = (d∞ (·,x∞ ◦ ) ≤ 2n+1 ). Thus Condition 3.2.2 of Definition 3.2.1 is also verified for the sequence (Bn)n=1,2,.... All the conditions in Definition 3.2.1 have been verified for the sequence ξ∞ ≡ (Bn)n=1,2,... to be a binary approximation of (S∞,d∞) relative to x∞ ◦ . Moreover, ξ∞ ≡ (|Bn|)n=1,2,... = (|An+1 n+1|)n=1,2,... ≡ (κn+1 n+1 )n=1,2,.... The lemma is proved. 3.3 Partition of Unity In this section, we define and construct a partition of unity relative to a binary approximation of a locally compact metric space (S,d). There are many different versions of partitions of unity in the mathematics liter- ature, providing approximate linear bases in the analysis of various linear spaces of functions. The present version, roughly speaking, furnishes an approximate linear basis for C(S,d), the space of continuous functions with compact supports on a locally compact metric space. In this version, the basis functions will be endowed with specific properties that make later applications simpler. For example, each basis function will be Lipschitz continuous. First we prove an elementary lemma for Lipschitz continuous functions. Lemma 3.3.1. Definition and basics for Lipschitz continuous functions. Let (S,d) be an arbitrary metric space. A real-valued function f on S is said to be Lipschitz continuous, with Lipschitz constant c ≥ 0, if |f (x) − f (y)| ≤ cd(x,y) for each x,y ∈ S. We will then also say that the function has Lipschitz constant c. Let x◦ ∈ S be an arbitrary but fixed reference point. Let f,g be real-valued functions with Lipschitz constants a,b, respectively, on S. Then the following conditions hold: 1. d(·,x◦) has Lipschitz constant 1. 2. αf +βg has Lipschitz constant |α|a +|β|b for each α,β ∈ R. If, in addition, |f | ≤ 1 and |g| ≤ 1, then fg has Lipschitz constant a + b. 3. f ∨ g and f ∧ g have Lipschitz constant a ∨ b. 4. 1 ∧ (1 − cd(·,x◦))+ has Lipschitz constant c for each c 0. 5. If f ∨ g ≤ 1, then fg has Lipschitz constant a + b. 6. Suppose (S ,d ) is a locally compact metric space. Suppose f is a real- valued function on S , with Lipschitz constant a 0. Suppose f ∨ f ≤ 1.
- 48. 28 Introduction and Preliminaries Then f ⊗f : S ×S → R has Lipschitz constant a +a , where S ×S is equipped with the product metric d ≡ d ⊗ d , and where f ⊗ f (x,x ) ≡ f (x)f (x ) for each (x,x ) ∈ S × S . 7. Assertion 6 can be generalized to a p-fold product f ⊗ f ⊗ · · · ⊗ f (p). Proof. Let x,y ∈ S be arbitrary. 1. By the triangle inequality, we have |d(x,x◦) − d(y,x◦)| ≤ d(x,y). Assertion 1 follows. 2. Note that |αf (x) + βg(x) − (αf (y) + βg(y))| ≤ |α(f (x) − f (y))| + |β(g(x) − g(y))| ≤ (|α|a + |β|b)d(x,y) and that if |f | ≤ 1 and |g| ≤ 1, then |f (x)g(x) − f (y)g(y)| ≤ |(f (x) − f (y))g(x)| + |(g(x) − g(y))f (y)|. ≤ |f (x) − f (y)| + |g(x) − g(y)| ≤ (a + b)d(x,y). Assertion 2 is proved. 3. To prove Assertion 3, first consider arbitrary r,s,t,u ∈ R. We will show that α ≡ |r ∨ s − t ∨ u| ≤ β ≡ |r − t| ∨ |s − u|. (3.3.1) To see this, we may assume, without loss of generality, that r ≥ s. If t ≥ u, then α = |r − t| ≤ β. On the other hand, if t ≤ u, then −β ≤ −|s − u| ≤ s − u ≤ r − u ≤ r − t ≤ |r − t| ≤ β, whence α = |r − u| ≤ β. By continuity, we see that inequality 3.3.1 holds for each r,s,t,u ∈ R. Consequently, |f (x) ∨ g(x) − f (y) ∨ g(y)| ≤ |f (x) − f (y)| ∨ |g(x) − g(y)| ≤ (a ∨ b)d(x,y) for each x,y ∈ S. Thus f ∨ g has Lipschitz constant a ∨ b. Assertion 2 then implies that the function f ∧ g = −((−f ) ∨ (−g)) also has Lipschitz constant a ∨ b. Assertion 3 is proved. 4. Assertion 4 follows immediately from Assertions 1, 2, and 3. 5. Assertion 5 follows from |f (x)g(x) − f (y)g(y)| ≤ |f (x)(g(x) − g(y))| + |(f (x) − f (y))g(y)| ≤ (b + a)d(x,y). 6. Suppose f ∨ f ≤ 1. Then for each (x,x ),(y,y ) ∈ S × S . |f (x)f (x ) − f (y)f (y )| ≤ |f (x)(f (x ) − f (y ))| + |(f (x) − f (y))f (y )| ≤ (a + a)(d(x ,y ) ∨ d(x,y)) ≡ (a + a)d ⊗ d ((x,x )),d(y,y ))
- 49. Partition of Unity 29 Thus the function f ⊗f : S ×S → R has Lipschitz constant a +a . Assertion 6 follows. 7. The proof of Assertion 7 is omitted. The next definitions and propositions embellish proposition 6.15 on page 119 of [Bishop and Bridges 1985]. Definition 3.3.2. ε-Partition of unity. Let A be an arbitrary metrically discrete and finite subset of a locally compact metric space (S,d). Because the set A is finite, we can write A = {x1, . . . ,xκ} for some sequence x ≡ (x1, . . . ,xκ), where x : {1, . . . ,κ} → A is an enumeration of the finite set A. Thus |A| ≡ κ. Let ε 0 be arbitrary. Define, for each k = 1, . . . ,κ, the function ηk ≡ 1 ∧ (2 − ε−1 d(·,xk))+ ∈ C(S,d) (3.3.2) and g+ k ≡ η1 ∨ · · · ∨ ηk ∈ C(S,d). (3.3.3) In addition, define g+ 0 ≡ 0. Also, for each k = 1, . . . ,κ, define gx(k) ≡ g+ k − g+ k−1. (3.3.4) Then the subset {gy : y ∈ A} of C(S,d) is called the ε-partition of unity of (S,d), determined by the enumerated set A. The members of {gy : y ∈ A} are called the basis functions of the ε-partition of unity. Proposition 3.3.3. Properties of ε-partition of unity. Let A = {x1, . . . ,xκ} be an arbitrary metrically discrete and enumerated finite subset of a locally compact metric space (S,d). Let ε 0 be arbitrary. Let {gx : x ∈ A} be the ε-partition of unity determined by the enumerated set A. Then the following conditions hold: 1. gx has values in [0,1] and has the set (d(·,x) 2ε) as support, for each x ∈ A. 2. x∈A gx ≤ 1 on S. 3. x∈A gx = 1 on x∈A(d(·,x) ≤ ε). 4. For each x ∈ A, the functions gx, y∈A;yx gy, and y∈A gy have Lipschitz constant 2ε−1. Here y x means y = xi and x = xj for some i,j ∈ {1, . . . ,κ} with i j. Proof. 1. Use the notations in Definition 3.3.2. Let k = 1, . . . ,κ be arbitrary. Suppose y ∈ S is such that gx(k)(y) 0. By the defining equality 3.3.4, it follows that g+ k (y) g+ k−1(y). Hence ηk(y) 0 by equality 3.3.3. Equality 3.3.2 then implies that d(y,xk) 2ε. Thus we see that the function gx(k) has (d(·,xk) 2ε) as support. In general, gx(k) has values in [0,1], thanks to equalities 3.3.2, 3.3.3, and 3.3.4. Assertion 1 is proved. 2. Note that x∈A gx = g+ κ ≡ η1 ∨ · · · ∨ ηκ ≤ 1. Assertion 2 is verified. 3. Suppose y ∈ S is such that d(y,xk) ≤ ε for some k = 1, . . . ,κ. Then ηk(y) = 1 according to equality 3.3.2. Hence x∈A gx(y) = g+ k (y) = 1 by equality 3.3.3. Assertion 3 is proved.
- 50. 30 Introduction and Preliminaries 4. Now let k = 1, . . . ,κ be arbitrary. Refer to Lemma 3.3.1 for the basic properties of Lipschitz constants. Then, in view of the defining equality 3.3.2, the function ηk has Lipschitz constant ε−1. Hence g+ k ≡ η0 ∨ · · · ∨ ηk has Lipschitz constant ε−1. In particular, y∈A gy ≡ g+ κ has Lipschitz constant ε−1. Moreover, for each k = 1, . . . ,κ, the function y∈A;yx(k) gy ≡ k−1 i=1 gx(i) = g+ k has Lipschitz constant ε−1, whence gx(k) ≡ g+ k − g+ k−1 has Lipschitz constant 2ε−1. Summing up, for each x ∈ A, the functions y∈A gy, y∈A;yx gy, and gx have Lipschitz constant c ≡ 2ε−1. Assertion 4 and the proposition are proved. Recall that if f ∈ C(S,d), then supx∈S |f (x)| exists and is denoted by f . Definition 3.3.4. Partition of unity of a locally compact metric space. Let (S,d) be a locally compact metric space, with a reference point x◦ ∈ S. Let the nondecreasing sequence ξ ≡ (An)n=1,2,... of enumerated finite subsets of (S,d) be a binary approximation of (S,d) relative to x◦. For each n ≥ 1, let {gn,x : x ∈ An} be the 2−n-partition of unity of (S,d) determined by the metrically discrete and finite subset An, as in Definition 3.3.2. Then the sequence π ≡ ({gn,x : x ∈ An})n=1,2,... is called a partition of unity of (S,d) determined by the binary approximation ξ. Proposition 3.3.5. Properties of partition of unity. Let ξ ≡ (An)n=1,2,... be a binary approximation of the locally compact metric space (S,d) relative to a reference point x◦. Let π ≡ ({gn,x : x ∈ An})n=1,2,... be the partition of unity determined by ξ. Let n ≥ 1 be arbitrary. Then the following conditions hold: 1. gn,x ∈ C(S,d) has values in [0,1] and has support (d(·,x) ≤ 2−n+1), for each x ∈ An. 2. x∈A(n) gn,x ≤ 1 on S. 3. x∈A(n) gn,x = 1 on x∈A(n)(d(·,x) ≤ 2−n). 4. For each x ∈ An, the functions gn,x, y∈A(n);yx gn,y, and y∈A(n) gn,y have Lipschitz constant 2n+1. Here y x means y = xi and x = xj for some i,j ∈ {1, . . . ,|An|} with i j. 5. For each x ∈ An, gn,x = y∈A(n+1) gn,xgn+1,y (3.3.5) on S. Proof. Assertions 1–4 are restatements of their counterparts in Proposition 3.3.3 for the case ε ≡ 2−n.
- 51. Partition of Unity 31 5. Now let x ∈ An be arbitrary. By Assertion 1, (gn,x 0) ⊂ (d(·,x) ≤ 2−n+1 ). At the same time, (d(·,x) ≤ 2−n+1 ) ⊂ (d(·,x◦) ≤ 2n+1 ) ⊂ y∈A(n+1) (d(·,y) ≤ 2−n−1 ) ⊂ ⎛ ⎝ y∈A(n+1) gn+1,y = 1 ⎞ ⎠, where the first inclusion is by Condition 3.2.2 of Definition 3.2.1, the second by Condition 3.2.1 of Definition 3.2.1 applied to n + 1, and the third by Assertion 3 applied to n + 1. Combining, (gn,x 0) ⊂ ⎛ ⎝ y∈A(n+1) gn+1,y = 1 ⎞ ⎠ . The desired equality 3.3.5 in Assertion 5 follows. The following proposition is a first application of partitions of unity. Proposition 3.3.6. Interpolation with linear combination of Lipschitz contin- uous functions. Let (S,d) be an arbitrary locally compact metric space, with an arbitrary but fixed reference point x◦. Let ξ ≡ (Ak)k=1,2,... be a binary approxi- mation of (S,d) relative x◦. Let π ≡ ({gk,x : x ∈ Ak})k=1,2,... be the partition of unity determined by ξ. Let n ≥ 1 be arbitrary but fixed. Let f ∈ C(Sn,dn) be arbitrary, with a modulus of continuity δf . Let ε 0 be arbitrary. Then the following conditions hold: 1. There exists k ≥ 1 so large that (i) the function f has the set B ≡ (d(·,x◦) ≤ 2k )n ⊂ Sn as support, and that (ii) 2−k 2−1δf (3−1ε). 2. Take any k ≥ 1 that satisfies Conditions (i) and (ii). Then sup (y,y ,...,y(n))∈Sn |f (y,y , . . . ,y(n) ) − (x,x ,...,x(n))∈A(k)n f (x,x , . . . ,x(n) )gx(y)gx (y ) · · · gx(n) (y(n) )| ≤ ε. (3.3.6) In other words, f − (x,x ,...,x(n))∈A(k)n f (x,x , . . . ,x(n) )gx ⊗ gx ⊗ · · · ⊗ gx(n) ≤ ε, (3.3.7) where · signifies the supremum norm in C(Sn,dn).
- 52. 32 Introduction and Preliminaries 3. The function represented by the sum in inequality 3.3.7 is Lipschitz continuous. 4. In particular, suppose n = 1 and |f | ≤ 1. Take any k ≥ 1 that satisfies Conditions (i) and (ii). Then f − g ≤ ε for some Lipschitz continuous function with Lipschitz constant 2k+1|Ak|. Proof. We will give the proof only for the case where n = 2; the other cases are similar. 1. Let K be a compact subset of C(S2,d2) such that the function f ∈ C(S2,d2) has K as support. Since the compact set K is bounded, there exists k ≥ 1 so large that K ⊂ B ≡ (d(·,x◦) ≤ 2k ) × (d(·,x◦) ≤ 2k ) and that (ii) 2−k 2−1δf (3−1ε). Conditions (i) and (ii) follow. Assertion 1 is proved. 2. Now fix any k ≥ 1 that satisfies Conditions (i) and (ii). Note that (d(·,x◦) ≤ 2k) ⊂ x∈A(k)(d(·,x) ≤ 2−k) by relation 3.2.1 of Definition 3.2.1. Hence Con- dition (i) implies that (i ) the function f has the set x∈A(k) (d(·,x) ≤ 2−k ) × x ∈A(k) (d(·,x ) ≤ 2−k ) as support. For abbreviation, write α ≡ 2−k, A ≡ Ak, and gx ≡ gk,x for each x ∈ A. 3. Let (y,y ) ∈ S2 be arbitrary. Suppose x,x ∈ A are such that gx(y)gx (y ) 0. Then, since gx,gx have (d(·,x) 2−k+1),(d(·,x ) 2−k+1), respectively, as support, according to Assertion 1 of Proposition 3.3.5, it follows that d2((y,y ), (x,x )) 2α δf (3−1ε). Consequently, the inequality |f (y,y ) − f (x,x )|gx(y)gx (y ) ≤ 3−1 εgx(y)gx (y ) (3.3.8) holds for arbitrary (x,x ) ∈ A2 such that gx(y)gx (y ) 0 and, consequently, holds for arbitrary (x,x ) ∈ A2. 4. There are two possibilities: (i ) |f (y,y )| 0 or (ii ) |f (y,y )| 3−1ε. 5. First consider case (i ). Then, by Condition (i ), we have (y,y ) ∈ (d(·,x◦) ≤ 2k ) × (d(·,x◦) ≤ 2k ) ⊂ B ≡ (x,x )∈A×A (d(·,x) 2−k ) × (d(·,x ) 2−k ). Hence, by Assertion 3 of Proposition 3.3.3, we have x∈A gx(y) = 1 and x ∈A gx (y ) = 1. Therefore
- 53. Partition of Unity 33 f (y,y ) − x∈A x ∈A f (x,x )gx(y)gx (y ) = x∈A x ∈A f (y,y )gx(y)gx (y ) − x∈A x ∈A f (x,x )gx(y)gx (y ) ≤ x∈A x ∈A |f (y,y ) − f (x,x )|gx(y)gx (y ) ≤ 3−1 ε x∈A x ∈A gx(y)gx (y ) ≤ 3−1 ε ε, where the second inequality follows from inequality 3.3.8, and where the third inequality is thanks to Assertion 2 of Proposition 3.3.3. 6. Now consider case (ii), where |f (y,y )| 3−1 ε. (3.3.9) Then f (y,y ) − x∈Ax ∈A f (x,x )gx(y)gx (y ) 3−1 ε + x∈Ax ∈A |f (x,x )|gx(y)gx (y ) ≤ 3−1 ε + x∈A x ∈A (|f (y,y )| + 3−1 ε)gx(y)gx (y ) ≤ 3−1 ε + x∈A x ∈A (3−1 ε + 3−1 ε)gx(y)gx (y ) ≤ 3−1 ε + (3−1 ε + 3−1 ε) = ε, (3.3.10) where the first and third inequalities are by inequality 3.3.9, where the second inequality is by inequality 3.3.8, and the last inequality is thanks to Assertion 2 of Proposition 3.3.3. 7. Summing up, Steps 5 and 6 show that f (y,y ) − x∈A x ∈A f (x,x )gx(y)gx (y ) ≤ ε, where (y,y ) ∈ S2 is arbitrary. The desired inequality 3.3.6 follows for the case where n = 2. The proof for the general case n ≥ 1 is similar. Assertion 2 of the present proposition is verified. 8. Let x,x ∈ A be arbitrary. Then the functions gx,gx are Lipschitz, according to Assertion 4 of Proposition 3.3.3. Hence the function gx ⊗ gx is Lipschitz, by Assertion 6 of Lemma 3.3.1. Assertion 3 of the present proposition follows. 9. Now suppose, in addition, that n = 1 and |f | ≤ 1. According to Proposition 3.3.3, each of the functions gx in the last sum has Lipschitz constant 2k+1, while f (x) is bounded by 1 by hypothesis. By Assertion 2, we have
- 54. 34 Introduction and Preliminaries f − x∈A(k) f (x)gx ≤ ε, (3.3.11) where the function g ≡ x∈A(k) f (x)gx has Lipschitz constant x∈A(k) |f (x)|2k+1 ≤ 2k+1 |Ak|, according to Assertion 2 of Lemma 3.3.1. Assertion 4 of the present proposition is proved. 3.4 One-Point Compactification The countable power of a locally compact metric space (S,d) is not necessarily locally compact, while the countable power of a compact metric space remains compact. For that reason, we will often find it convenient to embed a locally compact metric space into a compact metric space such that while the metric is not preserved, the continuous functions are. This embedding is made precise in the present section, by an application of partitions of unity. The next definition is an embellishment of definition 6.6, proposition 6.7, and theorem 6.8 of [Bishop and Bridges 1985]. Definition 3.4.1. One-point compactification. A one-point compactification of a locally compact metric space (S,d) is a compact metric space (S,d) with an element , called the point at infinity, such that the following five conditions hold: 1. d ≤ 1 and S ∪ {} is a dense subset of (S,d). 2. Let K be an arbitrary compact subset of (S,d). Then there exists c 0 such that d(x,) ≥ c for each x ∈ K. 3. Let K be an arbitrary compact subset of (S,d). Let ε 0 be arbitrary. Then there exists δK(ε) 0 such that for each y ∈ K and z ∈ S with d(y,z) δK(ε), we have d(y,z) ε. In particular, the identity mapping ῑ : (S,d) → (S,d), defined by ι(x) ≡ x for each x ∈ S, is uniformly continuous on each compact subset K of (S,d). 4. The identity mapping ι : (S,d) → (S,d), defined by ι(x) ≡ x for each x ∈ S, is uniformly continuous on (S,d). In other words, for each ε 0, there exists δd(ε) 0 such that d(x,y) ε for each x,y ∈ S with d(x,y) δd(ε). 5. For each n ≥ 1, we have (d(·,x◦) 2n+1 ) ⊂ (d(·,) ≤ 2−n ). Thus, as a point x ∈ S moves away from x◦ relative to d, it converges to the point at infinity relative to d. First we provide some convenient notations. Definition 3.4.2. Restriction of a family of functions. Let A,A be arbitrary sets and let B be an arbitrary subset of A. Recall that the restriction of a function
- 55. Discovering Diverse Content Through Random Scribd Documents
- 56. CHAPTER XXXIV.—A FIGHT WITH A HORNBILL. “Now me showee you something.” It was during the noon rest the next day and the Malay had asked the whites to come a little distance apart from the camp to a fine- looking banyan tree. They watched him with interest as with the axe he cut down several lengths of bamboo from a nearby cluster, and, pointing the ends sharply, having first separated the lengths into bits about two feet long, began driving them into the yielding bark of the tree. In this way he had soon made the first four rounds of a primitive ladder. Although, as yet, he had given them no hint of the object of all this, they were all sure that he had something really of interest to show them and forbore asking questions till he was ready to explain the mystery. Salloo had driven the tenth round of his queer ladder and was about ten feet from the ground, when Jack drew everybody’s attention to a strange hissing sound that appeared to come from within the tree. “Look out for snakes, Salloo,” he warned. But the Malay only nodded his head confidently and smiled. Donald glanced about nervously. Even Captain Sparhawk looked apprehensive. As for Muldoon, he shouted, “This is no place for a son of St. Patrick,” and fled back to camp. “What’s the matter, Salloo?” asked Mr. Jukes. “Are you in trouble?” “No trouble, Missel Boss,” rejoined Salloo. “Only bit what you callee good luck,” grinned the Malay, looking down on them and continuing his work. “How good luck?” asked Jack. “You see plenty soon,” was the cryptic reply, and the Malay drew another sharp-pointed peg from his girdle and drove it in with
- 57. vigorous strokes of the axe. While he did this, the hissing continued, mingled with a hoarse roaring like that which might be emitted by a disabled foghorn. Moreover, they could now see that a few feet above Salloo’s head was an object which alternately was thrust out from the tree trunk and withdrawn. It was white and sharp-pointed, like one of the pegs he was driving. It was assuredly not a snake’s head, as they had for a minute thought, but what was it? “What’s that right over your head, Salloo?” asked Captain Sparhawk. “Him buld (bird), captain. Him plentee much bigee buld.” “Oh, only a bird,” said Mr. Jukes in a disappointed voice. “What sort of a one?” “Him hornbill. Ole hen hornbill. She on nest. Old man hornbill he shut her up in there so she no leave eggs. Him put mud over crack in tree so as she no put nothing but her beak out. That the way he feedee her.” So that was the explanation of that object that darted in and out, and also of the hissing and grunting sounds. Looking closer, they now saw that at the spot where the bill still kept darting in and out there was a big longitudinal patch of mud which walled the hen hornbill up as effectually as certain prisoners were “walled up” in the days of old. As Salloo got within reaching distance of the nest, he raised his axe and smashed the mud wall before any of the party could check him. The next instant his bare arm was plunged fearlessly into the orifice and came out with his fingers clutching the old hen by the neck. In a moment she was fluttering, with her neck wrung, at the adventurers’ feet. “Say, Salloo, you shouldn’t have done that,” called up Jack indignantly. “That’s a shame.” The rest echoed his indignation at what seemed an act of wanton cruelty. Salloo only looked astonished. “Him plenty good eat. Roast hornbill plenty fine.”
- 58. “You see, he takes a different point of view about these things than we do,” said Captain Sparhawk. “You can’t blame him. Still I wish we could have prevented it.” They examined the dead hornbill with much interest. It was a gorgeous bird, almost as big as a turkey, with a bill of a size altogether disproportionate to even its large size. This beak was like a gigantic parrot’s bill and the horny structure extended over almost the entire head of the bird. It was not unlike the one the boys had shot the night before and thrown away as not good for food. “Plentee eggs in there,” said Salloo as he came down, “but they no good eat.” “Well, I’m glad there were no young ones to be starved through our interference,” said Billy, and the others felt as he did. “Say, I’m going to have a look at that nest,” said Jack suddenly. “All right. But look out you don’t fall and break your neck,” warned Raynor. Jack went nimbly up Salloo’s queer ladder and soon reached a height where he could see into the nest, which was built in a cavity of the tree and had afterward been carefully walled up with mud, strengthened by weaving reeds into it. Jack was still examining the nest when a sudden shadow fell over him. He looked up and above him he saw, with somewhat of a shock, a great bird whose plumage flashed brilliantly in the sun and whose huge beak snapped viciously at the boy. “Look out, look out, him father hornbill,” cried Salloo from the ground. The hornbill made a swoop at Jack, aiming with that cruel beak straight for his eyes. The boy put up an arm to defend himself, but the bird seized it with its parrot-like claws, scratching it badly, and all the while it kept up a beating of its wings that blinded the boy. Then the bird suddenly changed its tactics. It swooped off and then made a swift dash at the boy’s head. It was well for Jack that he had on his stiff sun helmet or his skull would have been cracked like an egg by that huge, horny bill. As it was, the helmet was ripped open.
- 59. Those below called on him to come down. But the attacks of the great bird so blinded and bewildered him that he was unable to move a step. Billy, at the order of Captain Sparhawk, brought a rifle from the camp, but so close did the bird stay to the boy that there was danger in using it. Even the most expert of shots would have been quite as likely to hit Jack as the enraged hornbill. Salloo had sprung into the tree, and with his ever ready kriss was ascending to the rescue when Captain Sparhawk saw an opportunity. The rifle was already at his shoulder and, as the hornbill rose and hovered for an instant before making another plunge at Jack’s head, his finger pressed the trigger. A splendid shot, a broken wing, the huge bird fluttered to the earth and flopped and screamed on the ground till its strugglings were put an end to by another bullet. Jack remained where he was for a few seconds to recover his nerves and then, still somewhat shaken by his experience, he descended. His arm was badly scratched and Captain Sparhawk was opening the medicine chest when Salloo intervened. He quickly gathered a handful of a plant that exuded a sort of thick milk. Crushing the gathered stems on a stone, he soon had a quantity of this juice, which he spread on the wounds. The irritation at once left them and Salloo promised a speedy cure. But it may be said that Jack had no appetite for roast hornbill that night.
- 61. CHAPTER XXXV.—THE HEART OF NEW GUINEA. The expedition now found itself advancing through forest that grew sparser as they progressed. The ground was rapidly becoming more rugged. Close to them now towered the range known as the Kini- Balu among the wild recesses of which the tribe of that name made its home. Constant vigilance was the watchword of the hour now. Salloo would permit no fires to be lighted, and he and his followers were constantly scouting in front of the party, while additional watch was kept at the rear and on both flanks. It was dangerous, thrilling work, but the boys, who loved adventure, relished every moment of it. But Donald Judson lived a life of misery. Every rustle in the bush made him turn pale. He was constantly giving false alarms in the night and the boys heartily wished he had been left behind. One afternoon—they were right in the mountains now—Salloo halted the party with a quick gesture. “Two men ahead of us. Up the mountain. Salloo go, look, see.” He glided off with his usual snake-like agility and vanished in a flash, while the party waited behind a mighty rock, for cover of the forest kind was growing scarce now. A wilder region would have been hard to imagine. The cliffs and mountains were of all sorts of extravagant shapes. Some of the larger rocks and peaks took on the outlines of monstrous animals. But they were still following a trail which was undoubtedly the one set down in red ink on Broom’s map. Through the glasses, which they were able to use without being observed, by crouching down in the coarse grass, they could see Salloo advancing toward the two figures on the mountain side. As he went he was making the peace sign, extending his arms as if inviting the others to attack him at their will. But as far as they could see, the meeting was friendly enough. Salloo conversed with the two
- 62. men of the mountain for a long time. Then he could be seen retracing his steps. “Well?” demanded everybody as he returned to the camp. “Ebblyting good so far,” reported Salloo. “Those two men velly old men. They left behind when tribe go to war in the north.” “Then the country is free of danger?” cried Donald. Salloo turned a look of contempt on him and did not answer. Addressing the others, he continued: “They say they know of cave. But no know if white man is there,” went on the Malay. “Would they be willing to guide us to it?” inquired Mr. Jukes. “That me no know yet. Me go see ’em again to-night,” replied Salloo. “They say nobody but old men, women and children left behind now tribe go to war. So maybe they no afraid to show us. You pay ’em good?” “Anything, any sum at all,” was the response of the millionaire. “No sum is too great to restore my brother to his family.” When night fell Salloo left the camp again and did not return till midnight. He brought the news that the two old men would guide them for three pieces of gold each. They did not want the coins to spend, explained Salloo, but to pierce and wear round their necks as ornaments. “I’ll make it six each,” declared Mr. Jukes, “if they lead us aright.” There was little sleep for anybody that night, and soon after daybreak the two old men appeared in the camp. They were odd- looking old fellows; unclothed except for a breech cloth, and were daubed with red and yellow earth, signifying that their tribe was at war, although their age barred them from taking part. At Salloo’s suggestion, only himself, Mr. Jukes, Jack and Billy were to accompany the guides. The others were to remain behind and keep as well under cover as they could till the rest returned with success
- 63. or failure. Final instructions having been given, they set off behind the two old men, who chattered volubly with Salloo as they went. They knew of the cave, it appeared, but nothing more, for they did not come from that part of the mountains. The next day they were not far from the cave, their aged guides told them, and Salloo enjoined the strictest caution in proceeding. If they met a returning war party, their position would be ticklish in the extreme, he declared, and they readily agreed with him. It was not long after this that, high up on the mountain side, they became aware of a dark hole. The two old men chattered and pointed, and then Salloo said: “There him cave. You wait here. Salloo go, look, see.” He made off up the mountain with the two old tribesmen, while the others waited with what patience they could for his return. The boys had never seen Mr. Jukes so nervous. He could not keep still under the tension, but paced to and fro, regardless of Salloo’s advice to keep under cover. “He is taking his time,” said Jack after a long interval. “Perhaps something has happened to him,” said Mr. Jukes, apprehensively. “We’d better have our pistols ready. Hark! what was that?” There was a rustling in the bushes near at hand and they all sprang to their feet, only to burst into laughter a minute later when a rock coney, or small rabbit, emerged, looked at them for an instant and then vanished. “That shows how we are keyed up,” said Jack. “We’ve got to keep our nerve or we shall be useless if any emergency did happen.” As he spoke, something whizzed over their heads and then sank quivering in the ground not far from them. They looked round and saw standing not far off two hideous natives, with frizzed hair and painted faces and bodies. Both were wounded and apparently had
- 64. been sent back from “the front.” But still there was a chance that they might be the advance guard of a big body of troops. “We friends,” cried Jack, giving the peace sign as he had seen Salloo give it. The natives merely stared, and there is no knowing what might have been the outcome, but at that moment there came a hail from high up on the mountain and the old tribesmen and Salloo began coming toward them. The natives awaited their coming with their eyes fixed on the whites. As soon as Salloo and the others arrived there was a long confab and Salloo explained that the two warriors said that the main body of the savage troops was not far off, and that they had been sent back on account of their wounds. They had thrown the spear because they thought the whites were coming to invade their country. When Salloo explained the object of their errand, everything appeared to be satisfactory. “Now we go to the cave,” said Salloo, at the end of these negotiations. “Him velly big one, me think.” “Did you—did you see any trace of my brother?” asked Mr. Jukes anxiously. “Me no see anything yet,” was the reply. “Me only go little way into cave.” “Then come, let us start at once,” said Mr. Jukes, stepping nimbly over the rough ground, in spite of his cumbersome build. As Salloo had said, the cave was a large one. It ran back fully a mile under the mountain. But they paid little attention to its natural beauties, so eager were they to find some trace of Jerushah Jukes. To one side was a swiftly flowing stream. They did not doubt that it came from a waterfall, the noise of which they could hear in the distance. Before long they stood in front of the waterfall, a beautiful ribbon of water falling fully a hundred feet into a clear pool. A sort of mist hung over the pool caused by the spray, which was lighted by a rift
- 65. in the rocks above. It was a lovely sight and even in their anxiety to get on they could not help standing and admiring it for a few minutes. “By the way, Salloo,” said Jack abruptly, “how about that ghost that is supposed to haunt the cave?” “Me no know. Me——” “Look, look, the ghost!” cried Raynor suddenly. He pointed straight in front of him at the fall. “Great Scott!” exclaimed Jack as he too perceived an apparition that appeared to rise out of the waters. Salloo fell flat on his face in terror and so did the two old natives, who had been their guides. “Don’t talk nonsense,” said Mr. Jukes sharply. “I see nothing. I—for heaven’s sake!” Out of the mist of the pool he had seen advancing toward him as he stepped forward the gigantic form of a man. Then he glanced again. The ghost was Mr. Jukes himself, who certainly had nothing spiritualistic about him. The explanation of the queer sight struck the boys and the millionaire at the same instant. The sun, shining through the rift, was reflected upon the wet rock which in turn projected their figures against the watery mist that hung above the pool. “And so that’s the ghost that’s been scaring the natives to death,” said Jack. “Get up, Salloo, and I’ll show you how the trick is done.” After a brief demonstration the Malay was satisfied, but the two old men were unconvinced. They mumbled and were ill at ease till that part of the cave was left behind. “Hullo, here’s a path leading up past the waterfall,” cried Jack suddenly. “So there is. Let’s see where it goes,” cried Billy. They started up the slippery footway very slowly so as to avoid the consequences of a slip. As they went it grew lighter. They were coming to the upper
- 66. world once more. A minute later and they emerged upon a small plateau in the heart of the mountains. It was surrounded by steep precipices. In the centre stood a group of bamboo huts. At sight of the white men, several women and children set up a shrill cry. Suddenly above the hub-bub came a voice that brought a thrill to them all: “Has help come at last?” “Has help come at last?”
- 67. From behind one of the huts had stepped a tall, angular figure, wearing ragged white clothes and a battered sun helmet. Perched on his nose were a pair of huge horn-rimmed spectacles, a ragged, unkempt beard covered his face and his hair hung in matted locks about his shoulders. At the sight of him, Mr. Jukes gave a gasp and then a glad cry.
- 69. CHAPTER XXXVI.—FOUND AT LAST! “Oh, my brother,” cried Mr. Jukes, “I can hardly believe we’ve found you at last.” “Thank God! you have, Jacob,” returned the other fervently. “For a moment I thought that you were only one of the fantastic visions that have visited my brain lately.” “My poor brother,” exclaimed the millionaire, “but now thank heaven you are restored to your friends.” “But how did you ever find me? I never deemed it possible that rescuers could find their way to this place where that villain Broom, after stealing the pearl, marooned me.” “Ah, so the pearl is gone,—but never mind that now. I would not have given your life for an ocean-full of pearls,” declared the millionaire happily, “but I must introduce our friends who have shared with me the hardships of the trail.” The boys, and then Salloo, added their congratulations to Mr. Jukes, while the women and children gathered round and chattered frantically. It was plain that they objected to all this, yet did not see how to stop it. The white men’s weapons glinted menacingly and there were no warriors in the village. “And now let us hasten away from here,” said Jerushah Jukes. “The men are off on a fighting expedition and I might have escaped but without food or weapons I could never have made my way to the coast through the jungle. I suppose that is the reason they did not tie me up.” “Undoubtedly,” said the millionaire, “but I’m forgetting something,” and he doled out to the two old men a reward, much over what they had demanded. They chattered their thanks glibly, making all sorts of gesticulations of gratitude.
- 70. “It’s all like a dream to me so far,” said Jerushah Jukes, as they made their way back through the cave and past the “haunted” waterfall. “Broom sent me up here with a guard of his men. The tribe appeared to be friendly to him and agreed to keep me prisoner as long as he wished. But my poor crew? What has become of them?” “That we do not know yet,” said Mr. Jukes, “but we will talk later. I want to put all the distance I can between this tribe and our party as soon as we can. Those women will give the alarm although they dared not make an active protest.” But as they emerged from the cave they met with a rude shock. A party of warriors with frizzed hair and war-paint daubing their bodies barred the way. At first the tribesmen stood motionless with astonishment at the sight of a party of white men emerging from their secret cave. But the next instant they broke into a savage volley of shouts and yells and raised their spears and cruel-looking war clubs. “We have come too late, my poor brother,” groaned Mr. Jukes. But suddenly Salloo raised his voice. He spoke in tones of loud authority. The spears and clubs were lowered. He turned to Mr. Jukes and in a quick low voice said: “Give me um map. Quick, our lives depend on him.” The millionaire lost no time in producing ‘Bully’ Broom’s map. The most be-frizzed of the natives pored over it for several minutes. Then one of them said in fair English: “You come from Chief Broom; all right, you may go. He tell us to keep white man till he send for him. You show Broom’s map. He all right. Goo’ bye,” and the warriors went on. Thus by the clever Malay’s strategy he had told the warriors, who had returned unexpectedly, that the white men had been sent by ‘Bully’ Broom,—they were saved from disaster. But the tribesmen had demanded proof of Salloo’s story and, in the nick of time, he
- 71. had luckily thought of the map which satisfied their suspicions at once, for Broom was the only white man, except the prisoner, who had ever visited the secret cave. The return to the camp was made without incident and Jack, on reaching it, at once rigged up his wireless apparatus and flashed to the Sea Gypsy the glad news of the rescue of the millionaire’s brother. But, a few minutes later, he, in his turn, was receiving good tidings. Broom had returned to Bomobori and was arrested while he was recruiting a crew to make a dash into the jungle and intercept the Jukes’ party. He was apprehended while rowing ashore from a native craft. As the officers of the law seized him, he was seen to throw something into the water. One of the native oarsmen instantly dived after the object and succeeded in grabbing it before it reached the bottom. It proved to be the great pearl, “The Tear of the Sea.” And there was yet more intelligence of a kind to hearten them after all their tribulations in the wild jungles of New Guinea. The first officer of the Sea Gypsy, having received news of a mysterious schooner anchored in a cove up the coast, resolved to do a little amateur detective work. He found that she was none other than the famous South Sea Lass. Securing the co-operation of the authorities, the vessel was raided one night and her small crew easily overpowered. Then cries were heard from below and on the removal of the hatches the crew of the Centurion, or what remained of them—for five had died from privation—were discovered. They had refused to join Broom’s band and he was afraid to let them loose, so they had been confined in the almost unlivable hold ever since their capture. Since Broom’s arrest, the Australian authorities had cabled that he was wanted there for piracy and other crimes and he had been sent to Melbourne on a mail steamer. It may be added here that British justice was dealt out with a heavy hand to the ruffian and his many victims were fully avenged. His crew was tried and sentenced in Bomobori and all received heavy terms of
- 72. imprisonment. Thus were the South Seas rid of one of the chief of their many freebooters. The long march back to Bomobori was made without anything of particular interest occurring and one morning they stood on a rise overlooking the harbor. There lay the Sea Gypsy with the dear old Stars and Stripes flying, and the ship dressed in gay bunting; for by wireless Jack had notified those on board of the time of their arrival. A few hours more and they were among their friends again with their strange experiences behind them. As there was no reason for staying in Bomobori, except to take on board the survivors of the Centurion’s crew, the Sea Gypsy steamed out of the harbor the next day, being saluted as she went, a compliment which she returned with her rapid-fire gun. Watching them from the wharf were two figures. One a tall agile Malay, who, with tears in his eyes, watched the yacht till she was hull-down on the horizon. It was Salloo. He had been well rewarded for his services which indeed, as Mr. Jukes said, were beyond price; but, as he watched the departure of his white friends, his thoughts were only with them and not with what were, to him, the riches of a lifetime. The other watcher turned away with a sneer, jingling the money Mr. Jukes had left him in his pockets: “So I’ve got to stick round this hole till I can get a steamer home,” grumbled Donald Judson, for, as our readers will have guessed, it was he. “If it hadn’t been for those boys I might have gone home in comfort on the yacht. Well, maybe some day I’ll get even with them.” On the voyage home a stop was made at the Pamatou Islands; the glad news of the rescue had already been wirelessed home, and there was no great hurry except Mr. Jukes’ desire to get back to his business affairs after a romantic adventure he would never forget. As the Sea Gypsy dropped anchor in the well-known harbor, a fleet of canoes dashed out to welcome her, among them you may be sure
- 73. those of Anai and his friend, who wept tears of joy at seeing their white “chums” once more. Mr. Jukes, his speculative instinct once more in the ascendent, bought a large quantity of pearls on which he subsequently realized a good profit. “But we must hurry home,” he said one day. “My business will be going to rack and ruin without me and besides I’ve run out of dyspepsia pills. I only hope I didn’t ruin my digestion in the jungle.” And here the adventures of the Ocean Wireless Boys on the Pacific must be brought to a close, except that it might be mentioned that pretty Helen Dennis, whose father’s ship was in port on the return of the Sea Gypsy, now wears a very pretty locket, set with South Sea pearls—the gift of Jack Ready. And so, till we meet them in the next volume of this series, we will wish the lads and their friends good- bye. THE END
- 74. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE A Volume of Cheerfulness in Rhyme and Picture KINDERGARTEN LIMERICKS By FLORENCE E. SCOTT. Pictures by Arthur O. Scott with a Foreword by Lucy Wheelock The book contains a rhyme for every letter of the alphabet, each illustrated by a full page picture in colors. The verses appeal to the child’s sense of humor without being foolish or sensational, and will be welcomed by kindergartners for teaching rhythm in a most entertaining manner. Beautifully printed and bound. In attractive box. Price, Postpaid One Dollar.
- 75. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE FRANK ARMSTRONG SERIES By MATTHEW M. COLTON Cloth Bound. Illustrated. Price, 60c. per vol., postpaid Frank Armstrong’s Vacation How Frank’s summer experiences with his boy friends make him into a sturdy young athlete through swimming, boating and baseball contests, and a tramp through the Everglades, is the subject of this splendid story. Frank Armstrong at Queens We find among the jolly boys at Queen’s School, Frank, the student- athlete, Jimmy, the baseball enthusiast, and Lewis, the unconsciously-funny youth who furnishes comedy for every page that bears his name. Fall and winter sports between intensely rival school teams are expertly described. Frank Armstrong’s Second Term The gymnasium, the track and the field make the background for the stirring events of this volume, in which David, Jimmy, Lewis, the “Wee One” and the “Codfish” figure, while Frank “saves the day.” Frank Armstrong, Drop Kicker With the same persistent determination that won him success in swimming, running and baseball playing, Frank Armstrong acquired the art of “drop-kicking,” and the Queen’s football team profits thereby. Frank Armstrong, Captain of the Nine Exciting contests, unexpected emergencies, interesting incidents by land and water make this story of Frank Armstrong a strong tale of
- 76. school-life, athletic success, and loyal friendships. Frank Armstrong at College With the development of this series, the boy characters have developed until in this, the best story of all, they appear as typical college students, giving to each page the life and vigor of the true college spirit. Six of the best books of College Life Stories published. They accurately describe athletics from start to finish. Any book sent postpaid upon receipt of 60 cents, or we will send the six for $3.50. HURST COMPANY, Publishers, NEW YORK
- 77. OAKDALE ACADEMY SERIES Stories of Modern School Sports By MORGAN SCOTT. Cloth Bound. Illustrated. Price, 60c. per vol., postpaid BEN STONE AT OAKDALE. Under peculiarly trying circumstances Ben Stone wins his way at Oakdale Academy, and at the same time enlists our sympathy, interest and respect. Through the enmity of Bern Hayden, the loyalty of Roger Eliot and the clever work of the “Sleuth,” Ben is falsely accused, championed and vindicated. BOYS OF OAKDALE ACADEMY. “One thing I will claim, and that is that all Grants fight open and square and there never was a sneak among them.” It was Rodney Grant, of Texas, who made the claim to his friend, Ben Stone, and this story shows how he proved the truth of this statement in the face of apparent evidence to the contrary. RIVAL PITCHERS OF OAKDALE. Baseball is the main theme of this interesting narrative, and that means not only clear and clever descriptions of thrilling games, but an intimate acquaintance with the members of the teams who played them. The Oakdale Boys were ambitious and loyal, and some were even disgruntled and jealous, but earnest, persistent work won out. OAKDALE BOYS IN CAMP. The typical vacation is the one that means much freedom, little restriction, and immediate contact with “all outdoors.” These conditions prevailed in the summer camp of the Oakdale Boys and made it a scene of lively interest.
- 78. THE GREAT OAKDALE MYSTERY. The “Sleuth” scents a mystery! He “follows his nose.” The plot thickens! He makes deductions. There are surprises for the reader— and for the “Sleuth,” as well. NEW BOYS AT OAKDALE. A new element creeps into Oakdale with another year’s registration of students. The old and the new standards of conduct in and out of school meet, battle, and cause sweeping changes in the lives of several of the boys. Any volume sent postpaid upon receipt of price. HURST COMPANY—Publishers—NEW YORK
- 79. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE Log Cabin to White House Series LIVES OF CELEBRATED AMERICANS Cloth Bound. Illustrated. Price, 60c., per vol., postpaid FROM BOYHOOD TO MANHOOD (The Life of Benjamin Franklin). By Wm. M. Thayer. Benjamin Franklin was known in the scientific world for his inventions and discoveries, in the diplomatic world because of his statesmanship, and everywhere, because of his sound judgment, plain speaking, and consistent living. FROM FARM HOUSE TO WHITE HOUSE (The Life of George Washington). By Wm. M. Thayer. The story of the hatchet and other familiar incidents of the boyhood and young manhood of Washington are included in this book, as well as many less well-known accounts of his experiences as surveyor, soldier, emissary, leader, and first president of the United States. FROM LOG CABIN TO WHITE HOUSE (The Life of James A. Garfield). By Wm. M. Thayer. It was a long step from pioneer home in Ohio where James A. Garfield was born, to the White House in Washington, and that it was an interesting life-journey one cannot doubt who reads Mr. Thayer’s account of it. FROM PIONEER HOME TO WHITE HOUSE (The Life of Abraham Lincoln). By Wm. M. Thayer. No President was ever dearer to the hearts of his people than was homely, humorous “Honest Abe.”
- 80. To read of his mother, his early home, his efforts for an education, and his rise to prominence is to understand better his rare nature and practical wisdom. FROM RANCH TO WHITE HOUSE (The Life of Theodore Roosevelt). By Edward S. Ellis, A. M. Every boy and girl is more or less familiar with the experiences of Mr. Roosevelt as Colonel and President, but few of them know him as the boy and man of family and school circles and private citizenship. Mr. Ellis describes Theodore Roosevelt as a writer, a hunter, a fighter of “graft” at home and of Spaniards in Cuba, and a just and vigorous defender of right. FROM TANNERY TO WHITE HOUSE (The Life of Ulysses S. Grant). By Wm. M. Thayer. Perhaps General Grant is best known to boys and girls as the hero of the famous declaration: “I will fight it out on this line if it takes all summer.” We will mail any of the above books prepaid at 60 cents each or the six for $3.50. HURST COMPANY, Publishers, NEW YORK
- 81. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE REX KINGDON SERIES By GORDON BRADDOCK Cloth Bound. Illustrated. Price, 60c. per volume Rex Kingdon of Ridgewood High A new boy moves into town. Who is he? What can he do? Will he make one of the school teams? Is his friendship worth having? These are the queries of the Ridgewood High Students. The story is the answer. Rex Kingdon in the North Woods Rex and some of his Ridgewood friends establish a camp fire in the North Woods, and there mystery, jealousy, and rivalry enter to menace their safety, fire their interest and finally cement their friendship. Rex Kingdon at Walcott Hall Lively boarding school experiences make this the “best yet” of the Rex Kingdon series. Rex Kingdon Behind the Bat The title tells you what this story is; it is a rattling good story about baseball. Boys will like it. Gordon Braddock knows what Boys want and how to write it. These stories make the best reading you can procure. Any book sent upon receipt of 60 cents each, or we will send all four of them for $2.30. HURST COMPANY, Publishers, NEW YORK
- 82. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE NEW BOOKS ON THE WAR GREAT WAR SERIES By MAJOR SHERMAN CROCKETT Cloth Bound. Price, 50c. postpaid Two American Boys with the Allied Armies Two American Boys in the French War Trenches Two American Boys with the Dardanelles Battle Fleet The disastrous battle raging in Europe between Germany and Austria on one side and the Allied countries on the other, has created demand for literature on the subject. The American public to a large extent is ignorant of the exact locations of the fighting zones with its small towns and villages. Major Crockett, who is familiar with the present battle-fields, has undertaken to place before the American boy an interesting Series of War stories. Get these three books and keep up-to-date. We will send any book for 50c., or the three of them for $1.25. HURST COMPANY, Publishers, NEW YORK
- 83. HURST COMPANY’S BOOKS FOR YOUNG PEOPLE BOY SCOUT SERIES ENDORSED BY BOY SCOUT ORGANIZATIONS By LIEUT. HOWARD PAYSON Cloth. Illustrated. Price 50c. Each BOY SCOUTS OF THE EAGLE PATROL In this story, self-reliance and self-defense through organized athletics are emphasized. BOY SCOUTS ON THE RANGE Cow-punchers, Indians, the Arizona desert and the Harkness ranch figure in this tale of the Boy Scouts. BOY SCOUTS AND THE ARMY AIRSHIP The cleverness of one of the Scouts as an amateur inventor and the intrigues of his enemies to secure his inventions make a subject of breathless interest. BOY SCOUTS’ MOUNTAIN CAMP Just so often as the reader draws a relieved breath at the escape of the Scouts from imminent danger, he loses it again in the instinctive impression, which he shares with the boys, of impending peril. BOY SCOUTS FOR UNCLE SAM Patriotism is a vital principle in every Boy Scout organization, but few there are who have such an opportunity for its practical expression as comes to the members of the Eagle Patrol. BOY SCOUTS AT THE PANAMA CANAL Most timely is this authentic story of the “great ditch.” It is illustrated by photographs of the Canal in process of Building.
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