![Contents
Preface to the First Edition xiii
Preface to the Second Edition xvii
Preface to the Third Edition xix
List of Notations xxi
1 Measure Spaces 1
§0 Introduction ©... ee ee te ee 1
§1 Measure on ao-algebraof Sets... ee 3
[I] o-algebra of Sets... ee ee 3
[II] Limits of Sequences of Sets... 2... 2. ee ee ee 4
[I] Generation of o-algebras. 2 6. ee ee 6
[IV] Borel o-algebras 2. 1 ee ne 9
[V] Measure ona g-algebra. 2... eee ee 11
[VI] Measures of aSequence of Sets... 1. eeee 14
[VII] Measurable Space and Measure Space .............-2000- 17
[VIII] Measurable Mapping... 2.6... ee ee ee ee 19
[IX] Induction of Measure by Measurable Mapping .............- 22
§2 Outer Measures... 6 ee es 29
[1] Construction of Measure by Means of Outer Measure ........... 29
[II] Regular Outer Measures 2.0 33
[I] Metric Outer Measures 2. ee ee 35
[IV] Construction of Outer Measures... 2... ee ee ee 38
§3 Lebesgue MeasureonR.... 2... 0... ee 42
[I] Lebesgue Outer Measureon RR... 2.2... 2. ee 42
[II] Some Properties of the Lebesgue Measure Space ............0- 47
[II] Existence of Non-Lebesgue Measurable Sets ..............- 51
[IV] Regularity of Lebesgue Outer Measure ........... 0000006 53
[V] Lebesgue Inner MeasureonR .. 0... 0.0.0. ee cee ee eee 60
§4 Measurable Functions ............00
02.022 eee 72
[1] Measurability of Functions 2... 1... ee ee eee 72
[II] Operations with Measurable Functions .........02.0000
05% 76
[IIT] Equality Almost Everywhere 2.0... ee 81
[IV] Sequence of Measurable Functions .. 1... 2... 000 eee eee 82
vii](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-12-320.jpg)
![viii Contents
[V] Continuity and Borel and Lebesgue Measurability of FunctionsonR... 86
[VI] Cantor Ternary Set and Cantor-Lebesgue Function ............ 88
§5 Completion of Measure Space 2.0... 6. ee eee 99
[I] Complete Extension and Completion of a Measure Space ......... 99
[11] Completion of the Borel Measure Space to the Lebesgue Measure Space 102
§6 Convergence a.e. and Convergence in Measure ................ 104
[I] Convergence ae... ee ee ee 104
[II] Almost Uniform Convergence ...........0 02020000005 108
[Il] Convergence in Measure 2... 1 ee et ill
[IV] Cauchy Sequences in Convergence in Measure ............05 116
[V] Approximation by Step Functions and Continuous Functions ....... 119
2 The Lebesgue Integral 131
§7 Integration of Bounded Functions on Sets of Finite Measure.......... 131
[I] Integration of Simple Functions ... 2... 20.2... 02.2002
eee 131
[I] Integration of Bounded Functions on Sets of Finite Measure ....... 136
[IIT] Riemann Integrability 2.0...
ee een 145
§8 Integration of Nonnegative Functions... 2... ee
eee ee 159
[I] Lebesgue Integral of Nonnegative Functions .............2.004 159
[II] Monotone Convergence Theorem .............-.2-2045. 161
[1] Approximation of the Integral by Truncation .............00- 169
§9 Integration of Measurable Functions ................-2-2504- 177
[I] Lebesgue Integral of Measurable Functions... 2... 0.2... ee 177
[II] Convergence Theorems .. 1... ee 186
[I] Convergence Theorems under Convergence in Measure ......... 190
[IV] Approximation of the Integral by Truncation ............... 191
[V] Translation and Linear Transformation of the Lebesgue IntegralonR . . 196
[VI] Integration by Image Measure... 2... ee 201
§10 Signed Measures ... 2.2... 2. es 212
[T] Signed Measure Spaces .. 2... ce et 212
[II] Decomposition of Signed Measures .. 2... 0... 0.0.0
ee eee 218
[III] Integration on a Signed Measure Space... 2... 0. ee eee eee 227
§11 Absolute Continuity ofa Measure 2... 0. ee 235
[I] The Radon-Nikodym Derivative 2.2.1...
0... eee ee eee 235
[I] Absolute Continuity of a Signed Measure Relative to a Positive Measure 236
[III] Properties of the Radon-Nikodym Derivative ............... 247
3 Differentiation and Integration 257
§12 Monotone Functions and Functions of Bounded Variation ........... 257
[I] The Derivative... ee 257
[I] Differentiability of Monotone Functions ..............-... 263
[OI] Functions of Bounded Variation... . 1... .. eee ee eee 274
§13 Absolutely Continuous Functions .............2020
000005 283
[I] Absolute Continuity... ee ee ee eee 283
[II] Banach-Zarecki Criterion for Absolute Continuity ............ 286](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-13-320.jpg)
![Contents ix
[1] Singular Functions . 2... ee ee 289
[IV] Indefinite Integrals 6. ee 289
[V] Calculation of the Lebesgue Integral by Means of the Derivative ..... 300
[VI] Length of Rectifiable Curves .............2 02000000 311
§14 Convex Functions 2.0... te ee 323
[1] Continuity and Differentiability of a Convex Function ........... 323
[1] Monotonicity and Absolute Continuity of a Convex Function ....... 332
[Il] Jensen’s Inequality... 2. eee 335
4 The Classical Banach Spaces 339
§15 Normed Linear Spaces . 2.0... ee ee ee ee ee ee eee 339
[I] Banach Spaces 0. ee ee ee ee eee 339
[1] Banach Spaces on Re eee ee ee ee ees 342
[II] The Space of Continuous Functions C([a,b]) .......2-..00- 345
[IV] A Criterion for Completeness of a Normed Linear Space ........ 347
[V] HilbertSpaces 2... 1... ee 349
[VI] Bounded Linear Mappings of Normed Linear Spaces. .......... 350
[VII] Baire Category Theorem... 1... ee ee 360
[VI] Uniform Boundedness Theorems ............0022
000 363
[IX] Open Mapping Theorem... 2... 366
[X] Hahn-Banach Extension Theorems... 2.2.0.0... 0000000008 373
[XI] Semicontinuous Functions... ..........0.22.0-.2000200- 386
§16 The L? Spaces 2... ee ee ee ee ee es 392
[I] The £? Spaces for p€ (0,00) 2... ee eee eee 392
[II] The Linear Spaces £? for p €[1,00) ©... . 2... cee ee eee 395
[I] The L? Spaces for p €[1,c0) 2.2... . ceeee 400
[IV] The SpaceL® 2... ee 410
[V] The L? Spaces forp € (0,1)... 6... ee
ee ee 417
[VI] Extensions of Hilder’s Inequality .... 0.0.0...
0.02. ee eee 422
§17 Relation among the L? Spaces... 1... ec ee ee 429
[I] The Modified Z? Norms for L? Spaces with p € [l,oo] ......... 429
[II] Approximation by Continuous Functions ............200056 431
[Il] L? Spaces withpe€ (0,1].......----0---0 020000002 435
[IV] The £2? Spaces 2. ee ee ee 439
§18 Bounded Linear Functionals on the L? Spaces ......-....0-020056 448
[I] Bounded Linear Functionals Arising from Integration ........... 448
[11] Approximation by Simple Functions .. 2... 0.20... 0.0 e eee 451
[II] A Converse of Hélder’s Inequality... 2... ........0.2000- 453
[IV] Riesz Representation Theorem on the L? Spaces ............- 457
§19 Integration on Locally Compact Hausdorff Space ..............-- 465
[I] Continuous Functions on a Locally Compact Hausdorff Space ...... 465
[II] Borel and Radon Measures .... 0.2... ce eee eee 470
[I] Positive Linear Functionals on C,(X) 2... ee 475
[IV] Approximation by Continuous Functions .............0005 483
[V] Signed Radon Measures .. 20... ee ee en 487](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-14-320.jpg)
![[VI] The Dual Space of C(X) «0. ee
5 Extension of Additive Set Functions to Measures
§20 Extension of Additive Set Functions on an Algebra... 2.2... 0.0005
[I] Additive Set Function on an Algebra 2... ee
[1] Extension of an Additive Set Function on an Algebra to a Measure ... .
[1] Regularity of an Outer Measure Derived from a Countably Additive Set
Function onan Algebra 2... ee
[IV] Uniqueness of Extension of a Countably Additive Set Function on
an Algebra toa Measure... 2. ee
[V] Approximation to a o-algebra Generated by an Algebra .........
[VI] Outer Measure Based ona Measure... ...........222005
§21 Extension of Additive Set Functions ona Semialgebra 2... 2... ee
[I] Semialgebras of Sets 2 1. ee
[0] Additive Set Function on a Semialgebra 2... ee ee
[I] Outer Measures Based on Additive Set Functions on a Semialgebra . . .
§22 Lebesgue-Stieltjes Measure Spaces 2... ee
ee ee
[I] Lebesgue-Stieltjes Outer Measures . 2... 1 ee
[I] Regularity of the Lebesgue-Stieltjes Outer Measures ...........
[II] Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure . .
[IV] Decomposition of an Increasing Function... .........--045-
§23 Product Measure Spaces . 2... 0 ee es
[I] Existence and Uniqueness of Product Measure Spaces ...........
[II] Integration on Product Measure Space .. 6.0...
eee ee ee
[II] Completion of Product Measure Space... 0... eee
ee eee
[[V] Convolution of Functions 2... 20.0... ee
[V] Some Related Theorems ...........--..---.-.----.
6 Measure and Integration on the Euclidean Space
§24 Lebesgue Measure Space on the Euclidean Space ...........--05
[I] Lebesgue Outer Measure on the Euclidean Space... ..........
[I] Regularity Properties of Lebesgue Measure SpaceonR” .........
[1] Approximation by Continuous Functions .............2004
[IV] Lebesgue Measure Space on R” as the Completion of a Product
Measure Space... 0. ee ee ee ee
[V] Translation of the Lebesgue IntegralonR™ ...........-0005
[VI] Linear Transformation of the Lebesgue IntegralonR” ..........
§25 Differentiation on the Euclidean Space... 1. ee
ee ee ee
[I] The Lebesgue Differentiation Theoremon RR” ...............
[01] Differentiation of Set Functions with Respect to the Lebesgue Measure
[III] Differentiation of the Indefinite Integral... 2. .......02..4.
[IV] Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure
[V] Signed Borel Measureson R? 2... 0... ee
[VI] Differentiation of Borel Measures with Respect to the Lebesgue Measure
§26 Change of Variable of Integration on the Euclidean Space... .........
658
664
666
673](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-15-320.jpg)
![Contents
[I] Change of Variable of Integration by Differentiable Transformations
[I] Spherical CoordinatesinR® 2.2... .. cc ee
[II] Integration by Image Measure on Spherical Surfaces ...........
7 Hausdorff Measures on the Euclidean Space
§27 Hausdorff Measures .. 1... 0... ce ee ee ene
[I] Hausdorff Measures on R?. 2... ee
[11] Equivalent Definitions of Hausdorff Measure .............06
[10] Regularity of Hausdorff Measure... 1... 1. ee ee eee ee
[IV] Hausdorff Dimension ... 2... 20... 0.0.00.
§28 Transformations of Hausdorff Measures .........--...-2-00---
[1] Hausdorff Measure of Transformed Sets ........0..-
000 ere
[II] 1-dimensional Hausdorff Measure ..........0..00
022 eee
[I] Hausdorff Measure of Jordan Curves .............-00-22--
§29 Hausdorff Measures of Integral and Fractional Dimensions ..........
[1] Hausdorff Measure of Integral Dimension and Lebesgue Measure
[1] Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in R”
[I] Transformation of Hausdorff Measure of Integral Dimension. ..... .
[IV] Hausdorff Measure of Fractional Dimension ...........0-2--
A Digital Expansions of Real Numbers
[I] Existence of p-digital Expansion .... 2.2.2.2... ..0.0000005
[II] Uniqueness Question in p-digital Representation .............
[10] Cardinality of the Cantor Ternary Set... 0... . ee ee ee
B Measurability of Limits and Derivatives
[I] Borel Measurability of Limits of aFunction ...........-2005
[11] Borel Measurability of the Derivative of aFunction ............
C Lipschitz Condition and Bounded Derivative
D Uniform Integrability
[I] Uniform Integrability 2... 0. ee
[MI] Equi-integrability 2... eee
[1] Uniform Integrability on Finite Measure Spaces .............
E Product-measurability and Factor-measurability
[I] Product-measurability and Factor-measurability of aSet ..........
[II] Product-measurability and Factor-measurability ofa Function ......
F Functions of Bounded Oscillation
[]) Partition of Closed Boxesin R® 2.2... ee
[I] Bounded OscillationinR® ........0......22
00.0000 008
[1] Bounded Oscillation on Subsets... 2... ee te te es
[IV] Bounded Oscillation on 1-dimensional Closed Boxes ..........
[V] Bounded Oscillation and Measurability ..............0006
[VI] Evaluation of the Total Variation of an Absolutely Continuous Function .
731
737
799](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-16-320.jpg)
![Preface xV
space. Here Hélder’s inequality and Minkowski’s inequality are proved for p € (0, oo]. §15
introduces the Banach space and its dual. §16 treats L? spaces for p € [1, oo] as well as for
p € (O, 1). §17 treats relation among the L? spaces for different values of p. The £7 spaces
of sequences of numbers (a, : n € N) with }7,,cn lan|? < 00 is treated as a particular case
of L? spaces in which the underlying measure space is the counting measure space on the
set N of natural numbers. The Riesz representation theorem on the L? spaces is proved in
§18. §19 treats integration on a locally compact Hausdorff space. Urysohn’s Lemma on
the existence of a continuous function with compact support and partition of unity, Borel
and Radon measures, the Riesz representation theorem on the space of continuous functions
with compact support as well as Lusin’s theorem on approximation of a measurable function
by continuous functions are included here. (The placement of §19 in Chapter 4 is somewhat
arbitrary.)
Chapter 5 treats extension of additive set functions to measures. It starts with extension
of an additive set function on an algebra to a measure in §20 and completes the theory with
extension of an additive set function on a semialgebra to a measure in §21. (Semialgebra of
sets is an abstraction of the aggregate of left-open and right-closed boxes in the Euclidean
space R”. Its importance lies in the fact that the Cartesian product of finitely many al-
gebras and in particular o-algebras is in general not an algebra, but only a semialgebra.)
As an example of extending an additive set function on a semialgebra to a measure, the
Lebesgue-Stieltjes measure determined by a real-valued increasing function on R is treated
in §22, Theorems establishing the equivalence of the absolute continuity and singularity
of a Lebesgue-Stieltjes measure with respect to the Lebesgue measure with the absolute
continuity and singularity of the increasing function that determines the Lebesgue-Stieltjes
measure are proved. As a second example of extending an additive set function on a semi-
algebra to a measure, the product measure on the product of finitely many measure spaces
is included in §23. Tonelli’s theorem and Fubini’s theorem on the reduction of a multiple
integral to iterated integrals are found here.
Chapter 6 specializes in integration in the Lebesgue measure space on R”. In §24 the
Lebesgue measure on R” is constructed as an extension of the notion of volumes of boxes
in R” to Lebesgue measurable subsets of R”. Then it is shown that the Lebesgue measure
space on R” is the completion of the n-fold product of the Lebesgue measure space on R.
Regularity of the Lebesgue measure and in particular approximation of Lebesgue measurable
sets by open sets leads to approximation of the integral of a measurable function by that of a
continuous function. Translation invariance of the Lebesgue measure and integral and linear
transformation of the Lebesgue measure and integral are treated. §25 begins with the study
of the average function of a locally integrable function. Hardy-Littlewood maximal theorem
and Lebesgue differentiation theorem are presented. These are followed by differentiation
of a set function with respect to the Lebesgue measure, in particular differentiation of a
signed Borel measure with respect to the Lebesgue measure, and density of a Lebesgue
measurable set with respect to the Lebesgue measure. §26 treats change of variable of
integration by differentiable transformations.
Chapter 7 is an introduction to Hausdorff measures on R”. §27 defines s-dimensional
Hausdorff measures on R" for s € [0, 00) and the Hausdorff dimension of a subset of R”.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-20-320.jpg)
![Preface to the Second Edition
In this new edition all chapters have been revised and additional material have been incor-
porated although the framework and organization of the book are unchanged. Specifically
the following sections have been added:
§13 [VI] Length of Rectifiable Curves
§15 [VII] Baire Category Theorem
[VI] Uniform Boundedness Theorem
[IX] Open Mapping Theorem
[X] Hahn-Banach Extension Theorems
§16 weak convergence in L? spaces in [III] and [TV] of §16
the complete metric spaces L? for p € (0, 1) in [V] of §16
§19 [V] Signed Radon Measures
[VI] Dual Space of C(X)
§23 [IV.2] Convolution of L? Functions
[IV.3] Approximate Identity in Convolution Product
[IV.4] Approximate Identity Relative to Pointwise Convergence
Besides these topics there are additional theorems in sections: §1, §4, §5, §8, §10, §11,
§13, §15, §16, §17, §19, §20, §21, §23, §24, §25, and §27. Also 64 problems have been
added.
To use this book as a textbook, selection of the following sections for instance makes a
possible one-year course at the graduate level:
$1 to §13, §15([T] to [VI]), §16 to §21, §23(M] to [III])
It is my pleasure to thank Abel Klein for his helpful comments on the first edition of
this book.
J. Yeh
Corona del Mar, California
March, 2006
xvii](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-22-320.jpg)
![Preface to the Third Edition
In this edition several topics are added. Since these topics do not fit in single sections they
are presented as appendices. They are:
[A] Digital Expansions of Real Numbers
[B] Measurability of Limits and Derivatives
[C] Lipschitz Condition and Bounded Derivative
[D] Uniform Integrability
[E] Product-measurability and Factor-measurability
[F] Functions of Bounded Oscillation
In [B], we show that if the limit of a real-valued function on R exists then it is Borel-
measurable. In particular if a real-valued function is differentiable then the derivative is
Borel-measurable.
In [C], we show that if a real-valued function satisfies a Lipschitz condition on [a, b] C R
then it is differentiable a.e. on [a, b] and moreover the derivative is bounded on [a, 5].
In [D], we discuss uniform integrability and equi-integrability.
In [F], we define the notion of bounded oscillation for a real-valued function whose domain
of definition is a closed box in R”. We show that for the particular case n = 1 a function is
of bounded oscillation if and only if it is a function of bounded variation.
Also 93 problems have been added. There is now atotal of 394 problems.
Corona del Mar, California
August, 2013
xix](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-24-320.jpg)
![§1 Measure on a o-algebra of Sets 3
§1 Measure on a c-algebra of Sets
[I] o-algebra of Sets
Notations. We write N for both the sequence (1, 2, 3, .. .) andthe set {1, 2, 3, ...}. Whether
a sequenice or a set is meant by N should be clear from the context. Similarly we write Z
for both (0, 1, —1, 2, —2, ...) and {0, 1, -1, 2, —2, ...} and Z, for both (0, 1, 2,...) and
{0, 1,2,...}.
Definition 1.1. Let X be an arbitrary set. A collection A of subsets of X is called an algebra
(or a field) of subsets of X if it satisfies the following conditions:
1° XeQ,
2 AEM=> AEA,
3 A,BEA>AUBEA.
Lemma 1.2. If 2 is an algebra of subsets of a set X, then
(1) GEM,
(2) At,...,
An € A= iy Ae € A,
G3) A BeADBANBEA,
(4) At,...,
An € A= (fy
Ac € A,
6) A BeA>ABem.
Proof. (1) follows from 1° and 2° of Definition 1.1. (2) is by repeated application of 3°.
Since AN B = (A‘ U B°)*, (3) follows from 2° and 3°. (4) is by repeated application of
(3). For (5) note that A B = AN BS € BW by 2° and (3). wf
Definition 1.3. An algebra A of subsets of a set X is called a o-algebra (or a o-field) if it
satisfies the additional condition:
4° (An:n EN) CA Une An € A
Note that applying condition 4° to the sequence (A, B, 0, 8, ...), we obtain condition
3° in Definition 1.1, Thus 3° is implied by 4°. Observe also that if an algebra 2 is a finite
collection, then itis ao-algebra. This follows from the fact that when 2 is a finite collection
then a countable union of members of 2 is actually a finite union of members of 2 and this
finite union is a member of & by (2) of Lemma 1.2.
Lemma 1.4. If & is a o-algebra of subsets of a set X, then
(0) (An:n
EN CAD (ey An € A
Proof. Note that (en 4n = (Upen AS)°- By 2°, A$ € and by 4°, nen Ag €
Thus by 2°, we have (U,ey AS)" € 2.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-32-320.jpg)
![4 CHAPTER 1 Measure Spaces
Notations. For an arbitrary set X, let $3(X) be the collection of all subsets of X. Thus
A & §B(X) is equivalent
to A C X.
Example 1. For an arbitrary set X, ¥3(X) satisfies conditions 1° - 3° of Definition 1.1
and condition 4° of Definition 1.3 and therefore it is a o-algebra of subsets of X. It is the
greatest a-algebra of subsets of X in the sense that if 2l is a o-algebra of subsets of X and
if P(X) C A thenA = F(X).
Example 2. For an arbitrary set X, {4, X} is a o-algebra of subsets of X. It is the smallest
o-algebra of subsets of X in the sense that if 2% is a o-algebra of subsets of X and if
Ac {B, X} then A = {B, X}.
Example 3. In R?, let 9% be the collection of all rectangles of the type (a1, bi] x (a2, bz]
where —oo < a; < b; < oo fori = 1, 2 with the understanding that (a;, 00] = (a, 00).
Let & be the collection of all finite unions of members of $t. We have Ht C A since
every A € 9 is the union of finitely many, actually one, members of 9% so that A € .
We regard § as the union of 0 members of 9% so that @ € &. It is easily verified that
Mis an algebra of subsets of R?. However & is not a o-algebra. Consider for instance,
An = (n—4,n] x ©, 1] € CM forn EN. Then U),ey An is not a finite union
of
members of 9 and is thus not a member of 2.
[II] Limits of Sequences of Sets
Definition 1.5. Let (A, : n € N) be a sequence of subsets of a set X. We say that
(A, : n € N) is an increasing sequence and write An t if An C Anti forn € N. We say
that (A, : n € N) is a decreasing sequence and write A, | if An D> Anyi forn ENA
sequence (A, : n € N) is called a monotone sequence if it is either an increasing sequence
or a decreasing sequence. For an increasing sequence (A, :n € N), we define
qd) lim An = U» = {x € X:x € A, for somen € N}.
RE.
For a decreasing sequence (A, : n € N), we define
(2) bm An = f) An = {x €X:x € A, foreveryne N}.
neN
For a monotone sequence (A, :n € N), im, A, always exists although it may be 9.
If A, ¢, then jim An = @if and only
if A, = 9 for every n € N. If A, |, we may
have tm, An = @ even if A, # for every n € N. Consider for example X = R and
An = (0,2) forn € N. Then lim A, =. On the other
hand if A, = [0, }) forn e N
n00
then A, | and jim, A, = {0}.
In order to define a limit for an arbitrary sequence (A, : n € N) of subsets of a set X
we define first the limit inferior and the limit superior of a sequence.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-33-320.jpg)
![6 CHAPTER 1 Measure Spaces
then lim A, = @.
n-00
Example. Let X = R and let a sequence (A, : n € N) of subsets of R be defined
by Ai = [0,1], 43 = [0,3], 4s = [0, 5],-.- and Ao = [0, 21, Ay = [0,4], Ag =
[0, 6], .... Then lim inf An = {x € X: x € A,for all but finitely
many n € N} = {0} and
lim sup A, = {x € X : x € A, for infinitely many n € N} = [0, 00). Thus jim, An does
ny
notexist, The subsequence (Ap, : k € N) = (Aj, A3, As, .. .) is adecreasing sequence with
jimn An, = {0} and the subsequence (A,, : k € N) = (Az, Aq, Ag, ...) is an increasing
sequence with jim n An = [0, 00).
Theorem 1.9, Let 2 be a o-algebra of subsets of a set X. For every sequence (A, :n € N)
in &, the two setslimminf A, and lim supAy,, are in 2. So is im a An ifit exists.
noo
Proof. For every n € N, (yo,
Az € A by Lemma 1.4. Then nen Mon At € A by
4° of Definition 1.3. This shows that lim inf An € &. Similarly U,., Ak € A by 4°
of Definition 1.3. Then cn jon At "2eA. by Lemma 1.4. Thus limsup A, € &. If
00
lim A, exists,
then lim A, = liminf
A, € &. .
n—>0o noo noo
[III] Generation of o-algebras
Let A be an arbitrary set. If we select a set E, corresponding to each a ¢€ A, then we
call {E, : a € A} a collection of sets indexed by A. Usual examples of indexing set A
are for instance
N = {1, 2,3,...}, Z= (0, 1, —1,2, —2,...}, and Z, = {0,1,2,...}. An
arbitrary set A can serve as an indexing set.
Lemma 1.10. Let {2l, : a € A} be a collection of o-algebras of subsets of a set X where
A is an arbitrary indexing set. Then 1yc4 Ua is a a-algebra of subsets of X. Similarly if
{2y : @ € A} is an arbitrary collection of algebras of subsets of X, then yc4 Aa is an
algebra of subsets of X.
Proof. Let {M. : @ € A} be an arbitrary collection of o-algebras of subsets of X. Then
Nee A, is a collection of subsets of X. To show that it is a o-algebra we verify 1°, 2°,
and 3° in Definition 1.1 and 4° in Definition 1.3. Now X € 2%, for every a € A so that
X € (ea Me verifying 1°. To verify
2°, note that if E € (,c4 Me, then E € A, so that
E° € Ay for every a € A and then E° € (,-4 Ay. 3° is implied by 4°. To verify 4°, let
(E, > n EN) C Oye4 Me. Then
for every a € A, we have (E, : n € N) C Ay so that
Unen En € My. Then Unen E,€ Mees Mo. Ot
Theorem 1.11. Let € be an arbitrary collection of subsets of a set X. There exists the
smallest o-algebra Ag of subsets of X containing €, smallest in the sense that if U is a
o-algebra of subsets of X containing € then Ay C A. Similarly there exists the smallest
algebra of subsets of X containing €.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-35-320.jpg)
![§1 Measure on a o-algebra of Sets 9
Let & be the collection of subsets K of X of the type
(3) K =(CNA‘)UB,
where C € o(€) and B € o4(EM A). Observe that since B C A, the union in (3) is a
disjoint union. By (3), X € # and K is closed under countable unions. To show that & is
also closed under complementations, let K € & be as given by (3). Then
K°=XK =[(KNA)UA][(CNA)UB]
=[(XN AD) (CN AY] U(A B)
since XN A° > CN AC and A > B. But (XM A°) (CN.A*) = C*N A*. Therefore
K° =(C°NA)U(A BER.
Thus & is closed under complementations and is therefore a o-algebra of subsets of X.
Next, observe that for any K € as given by (3) we have KM A = B € og(EN A) 80
that RN A C o4(EM A). Thus to show (2) it suffices to show that o(E)N AC ANA.
Since & is a o-algebra of subsets of X, it remains only to show that€ Cc A. LetEe €
and write FE = (EN A°) U(EN A). Since EN A € o4(EN A), E is a subset of X of type
(3). Thus E € & and therefore € Cc &. This completes the proof.
[TV] Borel o-algebras
To fix our terminology let us review definitions of some topological concepts. Let X be a
set. A collection of subsets of X is called a topology on Xif it satisfies the following
axioms:
I ged,
0 Xe,
I {Eyg:a€ A}CO> Ube ED,
IV Ei,E,ED > E()EeD.
The pair (X, 9) is called a topological space. The members of 9 are called the open sets
of the topological space.
A subset E of X is called a closed set if its complement E° is an open set. Thus X is
both an open set and a closed and so is 9.
An arbitrary union of open sets is an open set and a finite intersection of open sets is an
open set. An arbitrary intersection of closed sets is a closed set and a finite union of closed
sets is a closed set.
The interior E° of a subset E of X is defined as the union of all open sets contained in
E. Thus it is the greatest open set contained in E.
The closure E of E is defined as the intersection of all closed sets containing E. It is
the smallest closed set containing E.
The boundary 0£ of
£ is defined by 2E = (E° Uu (E))".
A subset £ of X is called a compact set if for every collection 23 of open sets such that
EC Uves V there exists a finite subcollection {Vj, ..., Vw} such that EC Ura Va.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-38-320.jpg)
![§1 Measure on a o-algebra of Sets 11
a sequence (C,, : n € N) of closed sets such that E = nw Cy. If we let F, = Ula Ck,
then (F, : n € N) is an increasing sequence of closed sets and nen Fn = Une Cn = E-
Thus an F,-set is always the limit of an increasing sequence of closed sets.
[V] Measure on a c-algebra
Notations. Let R = {—oo} URU {co} and call it the extended real number system. We use
the alternate notation [—oo, co] for R also.
Definition 1.19. Let € be a collection of subsets of aset X. Let y be anonnegative extended
real-valued set function on €. We say that
(a) y is monotone on € if y(E1) < y(E2) for Ei, Ez € € such that E, C E2,
(b) y is additive on € if y(E1 U E2) = y(E1) + y(E2) for E1, Ez € € such that
E, NE. =@and
E, UE. € €,
(© y is finitely additive on € if y (U1 Ex) = Di y (En for every disjoint finite
sequence (Ex: k =1,...,n) in € such that Ufa E,e €,
(d) y is countably additive on € if vy (Uncx En) = Vnen ¥ (En) for every disjoint
sequence (Ey, :n € N) in € such that |),
cy En € €
(©) y is subadditive on € if y(E, U Eo) < y(E1) + y(E2) for E1, Eo € € such that
E,UE€&
(@ y is finitely subadditive on € if y (Uha Ex) < Di (Ex) for every finite sequence
(Ex :k =1,...,n) in € such that (Z_;
Ex € €,
(g) y is countably subadditive on € if y (Unen En) < Yyen Y (En) for every sequence
(En in €N) in € such that ncn En € €.
Note that in (c) while LJf_, Ex € € is required, it is not required that any of J2_1 Ex.
Ubi Et, ..., U2zi Ex be in €. Note also that (c) implies (b) and (f) implies (¢).
Observation 1.20. Let y be a nonnegative extended real-valued set function on a collection
€ of subsets ofa set X. Assume that # € € and y(@) = 0.
(a) If y is countably additive on €&, then it is finitely additive on €.
(b) If y is countably subadditive on €, then it is finitely subadditive on €.
Proof. Suppose y is countably additive on €. To show that it is finitely additive on
€, let (Ex : k = 1,...,m) be a disjoint finite sequence in € such that Ute Eye €.
Consider the infinite sequence (F, : k € N) in € defined by & = E, fork = 1,...,n
and F, = fork >n+1. Since# € €, (RH : k € N) is a disjoint sequence in € with
Uben Fe = Uf Ex € €. Thus by the countable additivity of y on € and by the fact that
y(B) = 0, we have y (Ut Et) = » (Uren Fe) = Daew (Fe) = Cs y (Ex). This
proves the finite additivity of y on €. We show similarly that if y is countably subadditive
on &, then it is finitely subadditive on €.
Lemma 1.21, Let (£,, : n € N) be an arbitrary sequence in an algebra X of subsets of a
set X. Then there exists a disjoint sequence (F, : n € N) in Q such that](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-40-320.jpg)
![§1 Measure on a o-algebra of Sets 13
A with Ut_, Fe = Ufir Ex so that by the finite additivity and the monotonicity of y on
MA, we have y (Up_y Ex) = v (Utar Fe) = D1 v Fe) < D1 v (Ex). This proves the
finite subadditivity of y on 2f.
2. Suppose y is countably additive on 2{. To show that it is countable subadditive on
A, let (E, : n € N) be a sequence
in & such that ),.y En € UW. Let FP) = Ey and Fy =
E, (2. VU... E,-1) forn > 2. Then by Lemma 1.21, (F, : n € N) is a disjoint sequence
in & and en Fn = Unen En. Thus by the countable additivity and the monotonicity of
y on & by (a), we have ¥ (new En) = ¥ (nen Fn) = Lnen (Fn) = Donen ¥ (En).
This proves the countable subadditivity of y on 2.
Proposition 1.23. Let y be a nonnegative extended real-valued set function on an algebra
A of subsets of a set X. If y is additive and countably subadditive on A then y is countably
additive on A.
Proof. Suppose y is additive and countably subadditive on 2. To show that y is countably
additive on 2, let (E, : n € N) be a disjoint sequence in %& such that |),<y En € 2. The
additivity of y on 2% implies its monotonicity and finite additivity on 2by (a) of Lemma
1.22. Thus for every N ¢ N, wehave y(Unen En) = ¥(US, En) = 01 v(En). Since
this holds for every N € N, we have y(Unen En) = Donen ¥(En)- On the other hand,
by the countable subadditivity of y on &, we have y(U,ew En) < Lnen (En). Thus
v(Unen Ex) = Ynen (En). This proves the countable additivity of y on 2.
Definition 1.24. Let A be a o-algebra of subsets of a set X. A set function p defined on A
is called a measure if it satisfies the following conditions:
1° nonnegative extended real-valued: p(E) € [0, 00] for every E € &,
2° 4) = 0,
3° countable additivity: (En :n © N) CA, disjoint => u (Open En) = Cen H(En)-
Lemma 1.25. A measure u on a o-algebra A of subsets of a set X has the following
properties:
(1) finite additivity: (E,, ..., E,) C A, disjoint > w (Uta Ex) = Dee MED,
(2) monotonicity: E,, Ez € A, Ey C FE. > w(E1) < w(E2),
G) Ei, By € &, Ey C Ey, w(E1) < 00 = w(E2 Ei) = w(E2) — w(ED,
(4) countable subadditivity: (En :n € N) CM => w (hen En) < nen (En)
and in particular
(5) finite subadditivity: (E1,..., En) CM => w (Uta Ex) < DL w(Ed-
Proof. The countable additivity of ~ on A implies its finite additivity on 2% by (a) of
Observation 1.20. The finite additivity of jz on 2% implies its additivity on 2{ and then its
monotonicity on 2 by (a) of Lemma 1.22.
To prove (3), let £1, Hz € Wand Ey C Ey. Then £; and E> E; are two disjoint](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-42-320.jpg)
![14 CHAPTER 1 Measure Spaces
members of 2 whose union is equal to E27. Thus by the additivity of 2 on 2, we have
B(E2) = B(E}) + e(E2 £1). If w(E1) < 00, then subtracting (£1) for both sides of
the last equality, we have (£2) — (£1) = (E2 £1). This proves (3).
The countable additivity of jz on 2 implies its countable subadditivity on & by (b) of
Lemma 1.22. This then implies the finite subadditivity of 4. on 21 by (b) of Observation
1.20. &
Regarding (3) of Lemma 1.25, let us note that if 4(£,) = oo then by the monotonicity
of 2 we have (Ez) = 00 also so that 4.(£2) — w(E1) is not defined.
[VI] Measures of a Sequence of Sets
Let be a measure on a a-algebra & of subsets of a set X. Let (EZ, : n € N) be
a sequence in 2. If lim E&, exists, does lim y(E,) exist? If it does, do we have
n> noo
uf jim, En) = jim, uU(E,)? The next theorem addresses this question for monotone se-
quences of measurable sets. It is based on the countable additivity of a measure. It is a
fundamental theorem in that a subsequent theorem regarding the limit inferior and the limit
superior of the measures of an arbitrary sequence of measurable sets as well as the monotone
convergence theorem for the Lebesgue integral, Fatou’s lemma, and Lebesgue’s dominated
convergence theorem are ultimately based on this theorem.
Theorem 1.26. (Monotone Convergence Theorem for Sequences of Measurable Sets)
Let be a measure on a o-algebra A of subsets of a set X and let (E, :n € N) bea
monotone sequence in Q.
(a)
If En t, then lim j(En) = u4(lim Ey).
no 00
(b) FE, J, then lim p(E,) = a iim Ex), provided that there exists a set A € UA with
n—>00 > OO
HAA) < 00 such that Ey Cc A.
Proof. If Z,, +, then dim, En = Unen En € Wl. If Ey J, then dim, En = (hen En € A.
Note also that if (E, : n € N) is a monotone sequence in 2, then (u(Z,) : 2 € N) isa
monotone sequence in [0, oo] by the monotonicity of jz so that jim (En) exists in [0, oo].
1. Suppose E,, +. Then we have u(Z,) +. Consider first the case where 2(Ey,) = 00
for some mo € N. In this case we have Aim, w(En) = oo. Since Eno C Unen En =
lim E,, we have jt( lim E,) > (Eno) = 00. Thus j4( lim E,) =0co= lim (Ep).
noo noo ROO n> 00
Consider next the case where j4(E,) < 00 for every n € N. Let Ey = G and consider
a disjoint sequence (F, : n € N) in 2 defined by F,, = E, En-1 forn € N. We have
En = UN, Fa for every N € N and hence Unen En = Unen /n- Then we have
u( Jim En) = 2((JEn) = (Um) = a)
neN
neN neN
=oEn Env) =O {wEn) — o(En-v)},
neN ncN](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-43-320.jpg)
![16 CHAPTER 1 Measure Spaces
Remark 1.27. (b) of Theorem 1.26 has the following particular cases. Let (FE, : n € N) be
a decteasing sequence in %. Then lim (En) = a lim, E,) if any one of the following
conditions is satisfied:
(a) U(X) < 00,
(b) u(E1) < «0,
(c) 4(En,) < 00 for some no € N.
Proof. (a) and (b) are particular cases of (b) of Theorem 1.26 in which X and FE) respectively
are the containing set A € 21 with 4(A) < oo.
To prove (c), suppose 4(En,) < 00 for some no € N. Let (F;, : 2 € N) be a decreasing
sequence in 21 obtained by dropping the first no terms from (EZ, : n € N), that is, we set
F,, = Engin forn € N. Lemma 1.7 implies that lim inf i= lim inf E,, and lim sup F, =
n>00
lim sup EZ, and thus lim F, = lim E,. Now since (F, : n € N) is a decreasing sequence
noo noo n->00
and F, C En, forn € N and since 4(En,) < 00, (b) of Theorem 1.26 applies so that
lim »(F,) = u({ lim F,) = ( lim E,). Since (u(F,) : n € N) is a sequence obtained
noo noo noo
by dropping the first ng terms of (%(E,) : n € N), we have im, u(F,) = im, H(E,).
Therefore we have lim y(E,) = 2{ lim Ep).
a) n00
Let yz be a measure on a o-algebra A of subsets of a set X. Then for an arbitrary
sequence (E, :n € N) in 2, lim inf E,, and lim sup E,, exist
in 1 by Theorem 1.9 and thus
n>00
uf lim inf E,) and j.(lim sup E,,) are defined. Now (1(E,) : n € N) isasequencein [0, 00]
>
and thus liminf z(E,) = lim inf w(E,) and limsupu(E,) = lim sup 2(E,) exist in
n-00 n>ook>n n>0o N00 pon
[0, 00]. How arej4(liminf E,) and j:(lim sup E,) related respectively to lim inf 4(E,)
7100 noo 100
and lim sup 4(Z,)? The next theorem addresses this question.
ROO
Theorem 1.28. Let 2 be a measure on a o-algebra A of subsets of a set X.
(a) For an arbitrary sequence (E, :n € N) in Q, we have
() u(liminf £,) <lim inf CE).
n00 n>00
(b) If there exists A € SA with u(A) < 00 such that E, C Aforn €N, then
2) j(lim sup E,,) > lim sup 2(E,).
n->co n+>00
(c) both lim E, and lim p(E,) exist, then
ROO n->00
@) w( lim Eq) < tim (En).
noo n—>00
@ if jim En exist and if there exists A € A with w(A) < 00 such that E, C Aforn €N,
then im w(En) exists and
4 Hf lim En) = Jim w(En).](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-45-320.jpg)
![§1 Measure on a o-algebra of Sets 17
Proof. 1. Recall that lim inf E, = new Mon Be = tim, Chen Ex by the fact that
(Neen Ex : 2 € N) is an increasing sequence in %. Then by (a) of Theorem 1.26, we
have y(liminf
Zn) = lim (Myon Ee) = liminf (Myon Ze) since the limit of a
sequence, if it exists, is equal to the limit inferior of the sequence. Since hen Ex C En,
we have 14 (Msn Ex) < (En) for n € N by the monotonicity of 2. This then implies
lim:inf # (Men Ex) < lim inf u(E,). Continuing the chain of equalities above with this
inequality, we have (1).
2. Assume that there exists A € {2 with 4(A) < oo such that E, C A forn € N. Now
limsup En = nen Uson Ex = lim Ubon Ex by the fact that (Ubon Ex 2 €N) isa
n—00
decreasing sequence in A. Since E, C A for all m € N, we have Ubon E, Cc A for all
n & N. Thus we havejz(limsup Ep) = w( lim Uj, Ex) = lim 2 (Upon Ex) by 0)
n—>00 no9 RE n>00 I
of Theorem 1.26. Now lim (Uren 2x) = lim sup #(Usen Ex) since the limit of a se-
quence, if it exists, is equal to the limit superior of the sequence. Then by 5, Zz D En, we
have 1 (Ups, Ex) = w(En). Thus lim sup (Upon Ex) = lim sup u(E,). Continuing
the chain of equalities above with this jncquality, we have (2).
3. If im, E,, and jim, BA(E,) exist, then im, E,, = liminf E,, and im, BE) =
lim nf s.(i,) 80 that (1) reduces to (3) ue
“4. If ima, E,, exists, then im sup En =
= im, Ey,
= lim inf E,. If there exists
A € 2
with uA) < oo such that E, C "A for ne N.then by (2) and (1) we have
6) limsup 2(En) = #(lim sup E,} = u( lim Ep)
now Aco noo
= 2(liminf E,,) < liminf 2(E,).
n>00 n>00
Since liminf 4(Z,) < lim sup (£,) the inequalities (5) imply
noo n>00
(6) liminf (En) = w{ lim En) = lim sup HE).
Thus lim ,(£,) exists and then by (6) we have ( lim E,) = lim (Ey). This proves
(4) nora n->00 ni>00
[VII] Measurable Space and Measure Space
Definition 1.29. Let & be a o-algebra of subsets of a set X. The pair (X, QW) is called a
measurable space. A subset E of X is said to be &-measurable if E € A.
Definition 1.30. (a) if 4 is a measure on a o-algebra ‘XA of subsets of a set X, we call the
triple (X, A, w) a measure space.
(b) A measure pp on a o-algebra A of subsets of a set X is called a finite measure if](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-46-320.jpg)
![§1 Measure on a o-algebra of Sets 19
E with respect to us is a member of A (and consequently has (Eo) = 0 by the monotonicity
of 4). When & is complete with respect to 1, we say that (X, A, 4) is a complete measure
space,
Example. Let X = {a,b,c}. Then & = {9, {a}, {b, c}, X} is a o-algebra of subsets of
X. If we define a set function 4 on X by setting (0) = 0, w({a}) = 1, u({b,c}) = 0,
and 4.(X) = 1, then yw is a measure on 2. The set {b, c} is a null set in the measure space
(X, A, 2), but its subset {5} is not a member of 2. Therefore (X, A, 2) is not a complete
measure space.
Definition 1.35. (a) Given a measurable space (X, UM). An A-measurable set E is called
an atom of the measurable space if @ and E are the only A-measurable subsets of E.
(b) Given a measure space (X, A, 4). An A-measurable set E is called an atom of the
measure space if it satisfies the following conditions :
1° 2(E) > 0,
2 Eg C £, Eg € R= p(Eo) =0 or u(Eq) = (EZ).
Observe that if Z is an atom of (X, 20) and w(£) > 0, then E is an atom of (X, A, yz).
Example. In a measurable space (X, &) where X = {a, b, c} and A = {G, {a}, {b, c}, X},
if we define a set function on 2 by setting (8) = 0, w{{a}) = 1, u({b,c}) = 2, and
HX) = 3, then yz is a measure on &. The set {b, c} is an atom of the measure space
(X, 2, w).
[VI] Measurable Mapping
Let f be a mapping
of a subset D of a set X into
a set Y. We write D(f) and (f) for the
domain of definition and the range of f respectively. Thus
D(f=DcX,
MP) ={y oY:
y = f(&) forsomex e D(P)} CY.
For the image of D(f) by f we have f(D(f)) = N(f).
For an arbitrary subset E of Y we define the preimage of E under the mapping f by
f(E) = [x eX: f(x) € E} = {x DY): f@) € E}.
Note that F is an arbitrary subset of Y and need not be a subset of S8(f). Indeed E may
be disjoint from 9%(f), in which case f—1(E) = 9. In general we have f(f—1(E)) c E.
For an arbitrary collection € of subsets of Y, we let f-1(€) := {f-1(Z) : E € €}.
Observation 1.36. Given sets X and Y. Let f be a mapping with D(f) C X and](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-48-320.jpg)
![22 CHAPTER 1 Measure Spaces
Proof. Let V be an open set in Y. The continuity
of f on D implies that f—1(V) = UND
where U is an open set in X so that f—!(V) € %3y. Since this holds for every open set V
in Y, f is a 3x /My-measurable mapping of D into Y by (a) of Proposition 1.42.
A particular case of Theorem 1.43 is when we have a real-valued continuous function
f defined on a set D € By where By is the Borel o-algebra of subsets of a topological
space X. In this case we have (Y, By) = (R, BR). By Theorem 1.43, f is a Bx/Br-
measurable mapping of D into R.
[IX] Induction of Measure by Measurable Mapping
Let ys be a measure on a o-algebra 2 of subsets of a set X. We show next that a measurable
mapping of the measurable space (X, 2) into another measurable space (Y, 33) induces
a measure on the o-algebra 93. The induced measure on %B is called the image measure
induced by the measurable mapping.
Theorem 1.44. (Image Measure) Given two measurable spaces (X, A) and (Y, B). Let
f be aSA/B-measurable mapping of X into Y. Let be a measure on XU. The set function
defined by vy = 0 f—| on B, that is, v(B) = w (f-1(B)) for B € B, is a measure on
B.
Proof. Since f is a 21/93-measurable mapping of X into Y, we have f—1(B) € Ml forevery
B € ® and then v(B) = » (f—1(B)) € [0, oo]. Also vx) = u(f 1@) = w@ =0.
Let (B, : n € N) be a disjoint sequence in %3. Then (f—!(B,) : » € N) is a disjoint
sequence in & and f—! (U,c~ Bn) = Unew f1(Bn) € &. Thus we have the equality
»(Upen Ba) = #(F7"(Unen Bn) = Dnen #(F'(Bn)) = Caen ¥(Bn)- This shows
that v is countably additive on 1. Therefore v is a measure on 3.
Problems
Prob. 1.1. Given two sequences of subsets (E,, : n € N) and (F, : n € N) of aset X.
(a) Show that
(1) _sdiminf
E,, U liminf F, C liminf(E,
U F,) C liminf
E, Ulim sup Fy
noo AS>O
n>00 noo n-P0O
c lim sup(E£, U F,) C lim sup E,, U lim sup F,.
noo n>00 noo
(b) Show that
(2) liminf
E, M liminf F, C liminf(E,
N F,) C liminf
E, M lim sup F,
100 m0 100 100 no
C lim sup(E, N F,) C lim sup E, NM lim sup F,.
nooo n>00 noo](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-51-320.jpg)
![§1 Measure on a o-algebra of Sets 23
(c) Show thatif lim £, and lim F, exist, then lim (En u Fr) and lim (En n Fy) exist
n00 n00 n00 n00
and moreover
Q) lim (EZ,
U F,) = lim E,U lim Fy,
n-00 norco" n-+00
(4) lim (E, Fn) = lim E,A lim Fy.
n—->0O noo n—->00
Prob. 1.2. (a) Let (A, : n € N) be a sequence of subsets ofa set X. Let (B, :n € N) bea
sequence obtained by dropping finitely many entries in the sequence (A, : n € N). Show
that lim inf By =
= lim inf An and fim sup Ba
= im
m sup A,. Show that im, Bp, exists if
and onlyif jim, Ag exists andwhen ‘they exist they a
ae equal.
(b) Let (An:TR € N) and (B, : n € N) be two sequences of subsets of a set X such
that A, = 8B, for all but finitely many n € N. Show that fim inf By, = lim inf Ap and
lim sup B, = iimn sup Ay. Show that jim, B,, exists if and only’ if“iim A, exists and when
a>00
they exist they are
re equal.
Prob. 1.3. Let (Z,, : n € N) bea disjoint sequence of subsets of a set X. Show that im, En
exists and lim £, = 9.
n>00
Prob. 1.4. Leta € R and let (x, : n € N) be a sequence
of points in R, all distinct
from a, such that jim, X, = a. Show that im a {xn} exists and jima {xn} = 9 and thus
jim {xn} # {a}.
Prob. 1.5. ForE c Randt € R, let us write
E+¢ = {x +f €R:x € E} and call it
the translate of E by t. Let (f,: n € N) be a strictly decreasing sequence in R such that
jim, t, = Oand let E,= E +t, forn € N. Let us investigate the existence of im, En.
(a) Let E = (— 00, 0). Show that in, E, = (—00, 0].
(b) Let E = {a} where a € R. Show that Jim, E, =.
(c) Let
E = [a, b] where
a, b € Randa < b. “Show that fim, E, = (a, 6].
(d) Let E = (a, b) where
a, b € Randa < b. Show that jim, En
= (a, b].
(e) Let E = Q, the set of all rational numbers. Assume ‘that (i,: n € N) satisfies the
additional condition that £, is a rational number for all but finitely many n € N. Show that
Jim, E,= E.
Ol
Let E = Qas in (d) but assume that (1,: n € N) satisfies the additional condition that t,
is a rational number for infinitely many n € N and 1, is an irrational number for infinitely
many n € N. Show that Jima, E,, does not exist.
Prob. 1.6. The characteristic function 14 of a subset A of a set X is a function on X defined
by
1 forx € A,
1a@) = { 0 forx € A‘.
Let (A, : n € N) be a sequence of subsets of X and A be a subset of X.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-52-320.jpg)
![§1 Measure on a o-algebra of Sets 25
such that |_7_, Ay = X. Let ¥ be the collection of all arbitrary unions of members of €.
(a) Show that ¥ = o(€), the smallest o-algebra of subsets of X containing €.
(b) Show that the cardinality of o (€) is equal to 2".
Prob. 1.16. Let € = {A; : i € N} be a disjoint collection of nonempty subsets of a set X
such that |_),<9y Ai = X. Let ¥ be the collection of all arbitrary unions of members of €.
(a) Show that ¥ = o(€), the smallest o-algebra of subsets of X containing €.
(b) Show that the cardinality of o (€) is equal to 2%,
Prob. 1.17. Show that a o-algebra of subsets of a set cannot be a countably infinite collection,
that is, it is either a finite or an uncountable collection.
Prob. 1.18. Let € = {£1,--- , E,} be a finite collection of distinct, but not necessarily
disjoint, subsets of a set X. Let D be the collection of all subsets of X of the type:
Ay A
EL Ey? n+. Ee,
where A; assumes the values {1, 0} and E} = E; and E° = Ef fori = 1,...,n. Let
¥ be
the collection of all arbitrary unions of members of D.
(a) Show that any two distinct members of D are necessarily disjoint, that is, D is a disjoint
collection.
(b) Show that the cardinality of D is at most 2”.
(c) Show that
¥ = a(€).
(d) Show that the cardinality of #(€) has at most 2
(e) Show that o(€) = a(€).
Remark. For an arbitrary collection € of subsets of a set X, the smallest o-algebra of
subsets of X containing €, o(€), always exists according to Theorem 1.11. Prob. 1.18
presents a method of constructing a(€) for the case that € is a finite collection.
Prob. 1.19. Let € be an arbitrary collection of subsets of a set X. Consider a(€), the
smallest algebra of subsets of X containing €. Show that for every A € a(€) there exists
a finite subcollection €4 of € depending on A such that A € a(€,).
Prob. 1.20. Let € be an arbitrary collection of subsets of a set X. Consider o(€), the
smallest a-algebra of subsets of X containing €. Show that for every A € o(€) there
exists an at most countable subcollection €4 of € depending on A such that A € o(€,).
Prob. 1.21. Let yz be a measure on a o-algebra A of subsets of a set X and let Wy be a
sub-c-algebra of 2, that is, Xp is a o-algebra of subsets of X and Wo C MA. Show that the
restriction of jz to Ao is a measure on Wo.
Prob. 1.22. Let (X, A, 2) be a measure space. Show that for any £1, Ez € MU we have the
equality: (Ey U Ey) + w(E1 9 £2) = w(E1) + w(E2).
Prob. 1.23, Let (X, 2) be a measurable space. Let yzbe a measure on the a-algebra 2&
of subsets of X and let a, > 0 for every k € N. Define a set function yz on 2 by setting
= Deen Okie. Show that us is a measure on 2.
Prob. 1.24, Let X = (0, oc) and let J = {4 : k € N} where , = (K—1,k] fork EN.](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-54-320.jpg)
![This ebook is for the use of anyone anywhere in the United
States and most other parts of the world at no cost and with
almost no restrictions whatsoever. You may copy it, give it away
or re-use it under the terms of the Project Gutenberg License
included with this ebook or online at www.gutenberg.org. If you
are not located in the United States, you will have to check the
laws of the country where you are located before using this
eBook.
Title: Descriptive Analyses of Piano Works
Author: Edward Baxter Perry
Release date: February 14, 2014 [eBook #44910]
Most recently updated: October 24, 2024
Language: English
Credits: Produced by Sean (scribe_for_hire@yahoo.com), based
on
page images generously made available by the
Internet
Archive
(https://archive.org/details/descriptiveanaly00perriala).
*** START OF THE PROJECT GUTENBERG EBOOK DESCRIPTIVE
ANALYSES OF PIANO WORKS ***](https://image.slidesharecdn.com/17008090-250519152847-4bb70dc1/85/Real-Analysis-Theory-Of-Measure-And-Integration-3rd-Edition-J-Yeh-60-320.jpg)
Real Analysis Theory Of Measure And Integration 3rd Edition J Yeh
- 1. Real Analysis Theory Of Measure And Integration 3rd Edition J Yeh download https://ebookbell.com/product/real-analysis-theory-of-measure- and-integration-3rd-edition-j-yeh-34016180 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Real Analysis Theory Of Measure And Integration 2nd Edition J Yeh https://ebookbell.com/product/real-analysis-theory-of-measure-and- integration-2nd-edition-j-yeh-1007734 Problems And Proofs In Real Analysis Theory Of Measure And Integration 1st Edition James J Yeh https://ebookbell.com/product/problems-and-proofs-in-real-analysis- theory-of-measure-and-integration-1st-edition-james-j-yeh-49110644 Notes On Real Analysis And Measure Theory Fine Properties Of Real Sets And Functions Alexander Kharazishvili https://ebookbell.com/product/notes-on-real-analysis-and-measure- theory-fine-properties-of-real-sets-and-functions-alexander- kharazishvili-46249308 Real Analysis An Introduction To The Theory Of Real Functions And Integration 1st Edition Jewgeni H Dshalalow https://ebookbell.com/product/real-analysis-an-introduction-to-the- theory-of-real-functions-and-integration-1st-edition-jewgeni-h- dshalalow-2226844
- 3. Pattern Theory The Stochastic Analysis Of Realworld Signals Mumford D https://ebookbell.com/product/pattern-theory-the-stochastic-analysis- of-realworld-signals-mumford-d-2046222 Real Analysis Measure Theory Integration And Hilbert Spaces Elias M Stein Rami Shakarchi https://ebookbell.com/product/real-analysis-measure-theory- integration-and-hilbert-spaces-elias-m-stein-rami-shakarchi-51950918 Real Analysis Measure Theory Integration And Hilbert Spaces First Edition Elias M Stein Rami Shakarchi https://ebookbell.com/product/real-analysis-measure-theory- integration-and-hilbert-spaces-first-edition-elias-m-stein-rami- shakarchi-58228904 Real Analysis Measure Theory Integration And Hilbert Spaces Elias M Stein Rami Shakarchi https://ebookbell.com/product/real-analysis-measure-theory- integration-and-hilbert-spaces-elias-m-stein-rami-shakarchi-34751356 Real Analysis Measure Theory Integration And Hilbert Spaces Elias M Stein https://ebookbell.com/product/real-analysis-measure-theory- integration-and-hilbert-spaces-elias-m-stein-38429984
- 5. Sicsse Ty REAL ANALYSIS Theory of Measure and Integration J Yeh
- 6. REAL ANALYSIS Theory of Measure and Integration 3rd Edition
- 7. This page intentionally lett blank
- 8. 3rd Edition REAL ANALYSIS Theory of Measure and Integration J Yeh University of California, irvine Ye World Scientific NEW JERSEY » LONDON » SINGAPORE + BEIJING » SHANGHAI » HONG KONG » TAIPEI » CHENNAI
- 9. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Yeh, J. (James) Real analysis : theory of measure and integration / by J. Yeh, University of California, Irvine, USA. — 3rd edition. pages cm Includes bibliographical references and index. ISBN 978-981-4578-53-0 (hardcover : alk, paper) -- ISBN 978-981-4578-54-7 (pbk. : alk. paper) 1. Measure theory. 2. Lebesgue integral. 3. Integrals, Generalized. 4. Mathematical analysis. 5. Lp spaces. I. Title. QA312.¥44 2014 515'.42--1c23 2013049980 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved, This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by World Scientific Printers.
- 10. To my wife Betty
- 11. This page intentionally lett blank
- 12. Contents Preface to the First Edition xiii Preface to the Second Edition xvii Preface to the Third Edition xix List of Notations xxi 1 Measure Spaces 1 §0 Introduction ©... ee ee te ee 1 §1 Measure on ao-algebraof Sets... ee 3 [I] o-algebra of Sets... ee ee 3 [II] Limits of Sequences of Sets... 2... 2. ee ee ee 4 [I] Generation of o-algebras. 2 6. ee ee 6 [IV] Borel o-algebras 2. 1 ee ne 9 [V] Measure ona g-algebra. 2... eee ee 11 [VI] Measures of aSequence of Sets... 1. eeee 14 [VII] Measurable Space and Measure Space .............-2000- 17 [VIII] Measurable Mapping... 2.6... ee ee ee ee 19 [IX] Induction of Measure by Measurable Mapping .............- 22 §2 Outer Measures... 6 ee es 29 [1] Construction of Measure by Means of Outer Measure ........... 29 [II] Regular Outer Measures 2.0 33 [I] Metric Outer Measures 2. ee ee 35 [IV] Construction of Outer Measures... 2... ee ee ee 38 §3 Lebesgue MeasureonR.... 2... 0... ee 42 [I] Lebesgue Outer Measureon RR... 2.2... 2. ee 42 [II] Some Properties of the Lebesgue Measure Space ............0- 47 [II] Existence of Non-Lebesgue Measurable Sets ..............- 51 [IV] Regularity of Lebesgue Outer Measure ........... 0000006 53 [V] Lebesgue Inner MeasureonR .. 0... 0.0.0. ee cee ee eee 60 §4 Measurable Functions ............00 02.022 eee 72 [1] Measurability of Functions 2... 1... ee ee eee 72 [II] Operations with Measurable Functions .........02.0000 05% 76 [IIT] Equality Almost Everywhere 2.0... ee 81 [IV] Sequence of Measurable Functions .. 1... 2... 000 eee eee 82 vii
- 13. viii Contents [V] Continuity and Borel and Lebesgue Measurability of FunctionsonR... 86 [VI] Cantor Ternary Set and Cantor-Lebesgue Function ............ 88 §5 Completion of Measure Space 2.0... 6. ee eee 99 [I] Complete Extension and Completion of a Measure Space ......... 99 [11] Completion of the Borel Measure Space to the Lebesgue Measure Space 102 §6 Convergence a.e. and Convergence in Measure ................ 104 [I] Convergence ae... ee ee ee 104 [II] Almost Uniform Convergence ...........0 02020000005 108 [Il] Convergence in Measure 2... 1 ee et ill [IV] Cauchy Sequences in Convergence in Measure ............05 116 [V] Approximation by Step Functions and Continuous Functions ....... 119 2 The Lebesgue Integral 131 §7 Integration of Bounded Functions on Sets of Finite Measure.......... 131 [I] Integration of Simple Functions ... 2... 20.2... 02.2002 eee 131 [I] Integration of Bounded Functions on Sets of Finite Measure ....... 136 [IIT] Riemann Integrability 2.0... ee een 145 §8 Integration of Nonnegative Functions... 2... ee eee ee 159 [I] Lebesgue Integral of Nonnegative Functions .............2.004 159 [II] Monotone Convergence Theorem .............-.2-2045. 161 [1] Approximation of the Integral by Truncation .............00- 169 §9 Integration of Measurable Functions ................-2-2504- 177 [I] Lebesgue Integral of Measurable Functions... 2... 0.2... ee 177 [II] Convergence Theorems .. 1... ee 186 [I] Convergence Theorems under Convergence in Measure ......... 190 [IV] Approximation of the Integral by Truncation ............... 191 [V] Translation and Linear Transformation of the Lebesgue IntegralonR . . 196 [VI] Integration by Image Measure... 2... ee 201 §10 Signed Measures ... 2.2... 2. es 212 [T] Signed Measure Spaces .. 2... ce et 212 [II] Decomposition of Signed Measures .. 2... 0... 0.0.0 ee eee 218 [III] Integration on a Signed Measure Space... 2... 0. ee eee eee 227 §11 Absolute Continuity ofa Measure 2... 0. ee 235 [I] The Radon-Nikodym Derivative 2.2.1... 0... eee ee eee 235 [I] Absolute Continuity of a Signed Measure Relative to a Positive Measure 236 [III] Properties of the Radon-Nikodym Derivative ............... 247 3 Differentiation and Integration 257 §12 Monotone Functions and Functions of Bounded Variation ........... 257 [I] The Derivative... ee 257 [I] Differentiability of Monotone Functions ..............-... 263 [OI] Functions of Bounded Variation... . 1... .. eee ee eee 274 §13 Absolutely Continuous Functions .............2020 000005 283 [I] Absolute Continuity... ee ee ee eee 283 [II] Banach-Zarecki Criterion for Absolute Continuity ............ 286
- 14. Contents ix [1] Singular Functions . 2... ee ee 289 [IV] Indefinite Integrals 6. ee 289 [V] Calculation of the Lebesgue Integral by Means of the Derivative ..... 300 [VI] Length of Rectifiable Curves .............2 02000000 311 §14 Convex Functions 2.0... te ee 323 [1] Continuity and Differentiability of a Convex Function ........... 323 [1] Monotonicity and Absolute Continuity of a Convex Function ....... 332 [Il] Jensen’s Inequality... 2. eee 335 4 The Classical Banach Spaces 339 §15 Normed Linear Spaces . 2.0... ee ee ee ee ee ee eee 339 [I] Banach Spaces 0. ee ee ee ee eee 339 [1] Banach Spaces on Re eee ee ee ee ees 342 [II] The Space of Continuous Functions C([a,b]) .......2-..00- 345 [IV] A Criterion for Completeness of a Normed Linear Space ........ 347 [V] HilbertSpaces 2... 1... ee 349 [VI] Bounded Linear Mappings of Normed Linear Spaces. .......... 350 [VII] Baire Category Theorem... 1... ee ee 360 [VI] Uniform Boundedness Theorems ............0022 000 363 [IX] Open Mapping Theorem... 2... 366 [X] Hahn-Banach Extension Theorems... 2.2.0.0... 0000000008 373 [XI] Semicontinuous Functions... ..........0.22.0-.2000200- 386 §16 The L? Spaces 2... ee ee ee ee ee es 392 [I] The £? Spaces for p€ (0,00) 2... ee eee eee 392 [II] The Linear Spaces £? for p €[1,00) ©... . 2... cee ee eee 395 [I] The L? Spaces for p €[1,c0) 2.2... . ceeee 400 [IV] The SpaceL® 2... ee 410 [V] The L? Spaces forp € (0,1)... 6... ee ee ee 417 [VI] Extensions of Hilder’s Inequality .... 0.0.0... 0.02. ee eee 422 §17 Relation among the L? Spaces... 1... ec ee ee 429 [I] The Modified Z? Norms for L? Spaces with p € [l,oo] ......... 429 [II] Approximation by Continuous Functions ............200056 431 [Il] L? Spaces withpe€ (0,1].......----0---0 020000002 435 [IV] The £2? Spaces 2. ee ee ee 439 §18 Bounded Linear Functionals on the L? Spaces ......-....0-020056 448 [I] Bounded Linear Functionals Arising from Integration ........... 448 [11] Approximation by Simple Functions .. 2... 0.20... 0.0 e eee 451 [II] A Converse of Hélder’s Inequality... 2... ........0.2000- 453 [IV] Riesz Representation Theorem on the L? Spaces ............- 457 §19 Integration on Locally Compact Hausdorff Space ..............-- 465 [I] Continuous Functions on a Locally Compact Hausdorff Space ...... 465 [II] Borel and Radon Measures .... 0.2... ce eee eee 470 [I] Positive Linear Functionals on C,(X) 2... ee 475 [IV] Approximation by Continuous Functions .............0005 483 [V] Signed Radon Measures .. 20... ee ee en 487
- 15. [VI] The Dual Space of C(X) «0. ee 5 Extension of Additive Set Functions to Measures §20 Extension of Additive Set Functions on an Algebra... 2.2... 0.0005 [I] Additive Set Function on an Algebra 2... ee [1] Extension of an Additive Set Function on an Algebra to a Measure ... . [1] Regularity of an Outer Measure Derived from a Countably Additive Set Function onan Algebra 2... ee [IV] Uniqueness of Extension of a Countably Additive Set Function on an Algebra toa Measure... 2. ee [V] Approximation to a o-algebra Generated by an Algebra ......... [VI] Outer Measure Based ona Measure... ...........222005 §21 Extension of Additive Set Functions ona Semialgebra 2... 2... ee [I] Semialgebras of Sets 2 1. ee [0] Additive Set Function on a Semialgebra 2... ee ee [I] Outer Measures Based on Additive Set Functions on a Semialgebra . . . §22 Lebesgue-Stieltjes Measure Spaces 2... ee ee ee [I] Lebesgue-Stieltjes Outer Measures . 2... 1 ee [I] Regularity of the Lebesgue-Stieltjes Outer Measures ........... [II] Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure . . [IV] Decomposition of an Increasing Function... .........--045- §23 Product Measure Spaces . 2... 0 ee es [I] Existence and Uniqueness of Product Measure Spaces ........... [II] Integration on Product Measure Space .. 6.0... eee ee ee [II] Completion of Product Measure Space... 0... eee ee eee [[V] Convolution of Functions 2... 20.0... ee [V] Some Related Theorems ...........--..---.-.----. 6 Measure and Integration on the Euclidean Space §24 Lebesgue Measure Space on the Euclidean Space ...........--05 [I] Lebesgue Outer Measure on the Euclidean Space... .......... [I] Regularity Properties of Lebesgue Measure SpaceonR” ......... [1] Approximation by Continuous Functions .............2004 [IV] Lebesgue Measure Space on R” as the Completion of a Product Measure Space... 0. ee ee ee ee [V] Translation of the Lebesgue IntegralonR™ ...........-0005 [VI] Linear Transformation of the Lebesgue IntegralonR” .......... §25 Differentiation on the Euclidean Space... 1. ee ee ee ee [I] The Lebesgue Differentiation Theoremon RR” ............... [01] Differentiation of Set Functions with Respect to the Lebesgue Measure [III] Differentiation of the Indefinite Integral... 2. .......02..4. [IV] Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure [V] Signed Borel Measureson R? 2... 0... ee [VI] Differentiation of Borel Measures with Respect to the Lebesgue Measure §26 Change of Variable of Integration on the Euclidean Space... ......... 658 664 666 673
- 16. Contents [I] Change of Variable of Integration by Differentiable Transformations [I] Spherical CoordinatesinR® 2.2... .. cc ee [II] Integration by Image Measure on Spherical Surfaces ........... 7 Hausdorff Measures on the Euclidean Space §27 Hausdorff Measures .. 1... 0... ce ee ee ene [I] Hausdorff Measures on R?. 2... ee [11] Equivalent Definitions of Hausdorff Measure .............06 [10] Regularity of Hausdorff Measure... 1... 1. ee ee eee ee [IV] Hausdorff Dimension ... 2... 20... 0.0.00. §28 Transformations of Hausdorff Measures .........--...-2-00--- [1] Hausdorff Measure of Transformed Sets ........0..- 000 ere [II] 1-dimensional Hausdorff Measure ..........0..00 022 eee [I] Hausdorff Measure of Jordan Curves .............-00-22-- §29 Hausdorff Measures of Integral and Fractional Dimensions .......... [1] Hausdorff Measure of Integral Dimension and Lebesgue Measure [1] Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in R” [I] Transformation of Hausdorff Measure of Integral Dimension. ..... . [IV] Hausdorff Measure of Fractional Dimension ...........0-2-- A Digital Expansions of Real Numbers [I] Existence of p-digital Expansion .... 2.2.2.2... ..0.0000005 [II] Uniqueness Question in p-digital Representation ............. [10] Cardinality of the Cantor Ternary Set... 0... . ee ee ee B Measurability of Limits and Derivatives [I] Borel Measurability of Limits of aFunction ...........-2005 [11] Borel Measurability of the Derivative of aFunction ............ C Lipschitz Condition and Bounded Derivative D Uniform Integrability [I] Uniform Integrability 2... 0. ee [MI] Equi-integrability 2... eee [1] Uniform Integrability on Finite Measure Spaces ............. E Product-measurability and Factor-measurability [I] Product-measurability and Factor-measurability of aSet .......... [II] Product-measurability and Factor-measurability ofa Function ...... F Functions of Bounded Oscillation []) Partition of Closed Boxesin R® 2.2... ee [I] Bounded OscillationinR® ........0......22 00.0000 008 [1] Bounded Oscillation on Subsets... 2... ee te te es [IV] Bounded Oscillation on 1-dimensional Closed Boxes .......... [V] Bounded Oscillation and Measurability ..............0006 [VI] Evaluation of the Total Variation of an Absolutely Continuous Function . 731 737 799
- 18. Preface to the First Edition This monograph evolved froma set of lecture notes for a course entitled Real Analysis that I taught at the University of California, Irvine. The subject of this course is the theory of measure and integration. Its prerequisite is advanced calculus. All of the necessary background material can be found, for example, in R. C. Buck’s Advanced Calculus. The course is primarily for beginning graduate students in mathematics but the audience usually includes students from other disciplines too. The first five chapters of this book contain enough material for a one-year course. The remaining two chapters take an academic quarter to cover. Measure is a fundamental concept in mathematics. Measures are introduced to estimate sizes of sets. Then measures are used to define integrals. Here is an outline of the book. Chapter 1 introduces the concepts of measure and measurable function. §1 defines measure as a nonnegative countably additive set function on a o-algebra of subsets of an arbitrary set. Measurable mapping from a measure space into another is then defined. §2 presents construction of a measure space by means of an outer measure. To have a concrete example of a measure space early on, the Lebesgue measure space on the real line R is introduced in §3. Subsequent developments in the rest of Chapter 1 and Chapter 2 are in the setting of a general measure space. (This is from the consideration that in the definition of a Measure and an integral with respect to a measure the algebraic and topological structure of the underlying space is irrelevant and indeed unnecessary. Topology of the space on which a measure is defined becomes relevant when one considers the regularity of the measure, that is, approximation of measurable sets by Borel sets.) §4 treats measurable functions, in particular algebraic operations on measurable functions and pointwise limits of sequences of measurable functions. §5 shows that every measure space can be completed. §6 compares two modes of convergence of a sequence of measurable functions: convergence almost everywhere and convergence in measure. The Borel-Cantelli Lemma and its applications are presented. A unifying theorem (Theorem 6.5) is introduced from which many other convergence theorems relating the two modes of convergence are derived subsequently. These include Egoroff’s theorem on almost uniform convergence, Lebesgue’s and Riesz’s theorems. Chapter 2 treats integration of functions on an arbitrary measure space. In §7 the Lebesgue integral, that is, an integral with respect to a measure, is defined for a bounded. real-valued measurable function on a set of finite measure. The Bounded Convergence Theorem on the commutation of integration and limiting process for a uniformly bounded.
- 19. xiv Preface sequence of measurable functions which converges almost everywhere on a set of finite measure is proved here. The proof is based on Egoroff’s theorem. On the Lebesgue measure space on R, comparison of the Lebesgue integral and the Riemann integral is made. §8 contains the fundamental idea of integration with respect to a measure. It is shown here that for every nonnegative extended real-valued measurable function on a measurable set the integral with respect to the measure always exists even though it may not be finite. The Monotone Convergence Theorem for an increasing sequence of nonnegative measurable functions, the most fundamental of all convergence theorems regarding commutation of integration and convergence of the sequence of integrands, is proved here. Fatou’s Lemma concerning the limit inferior of a sequence of nonnegative measurable functions is derived from the Monotone Convergence Theorem. In §9 the integral of an extended real-valued measurable function on a measurable set is then defined as the difference of the integrals of the positive and negative parts of the function provided the difference exists in the extended real number system. The generalized monotone convergence theorem for a monotone sequence of extended real-valued measurable functions, generalized Fatou’s lemma for the limit inferior and the limit superior of a sequence of extended real-valued measurable functions, and Lebesgue’s Dominated Convergence Theorem are proved here. Fatou’s Lemma and Lebesgue’s Dominated Convergence Theorem under convergence in measure are included. In §10 a signed measure is defined as an extended real-valued countably additive set function on a o-algebra and then shown to be the difference of two positive measures. In §11 the Radon-Nikodym derivative of a signed measure with respect to a positive measure is defined as a function which we integrate with respect to the latter to obtain the former. The existence of the Radon-Nikodym derivative is then proved under the assumption that the former is absolutely continuous with respect to the latter and that both are o-finite. (The fact that the Radon-Nikodym derivative is a derivative not only in name but in fact it is the derivative of a measure with respect to another is shown for Borel measures on the Euclidean space in §25.) Chapter 3 treats the interplay between integration and differentiation on the Lebesgue measure space on R. §12 presents Lebesgue’s theorem that every real-valued increasing function on R is differentiable almost everywhere on R. The proof is based on a Vitali covering theorem. This is followed by Lebesgue’s theorem on the integral of the derivative of a real-valued increasing function on a finite closed interval in R. Functions of bounded variation are included here. §13 defines absolute continuity of a real-valued function on a finite closed interval in R and then shows that a function is absolutely continuous if and only if it is an indefinite integral of a Lebesgue integrable function. This is followed by Lebesgue’s decomposition of a real-valued increasing function as the sum of an absolutely continuous function and a singular function. Such methods of calculating a Riemann integral in calculus as the Fundamental Theorem of Calculus, integration by parts, and change of variable of integration find their counterparts in the Lebesgue integral here. §14 treats convex functions and in particular their differentiability and absolute continuity property. Jensen’s inequality is included here. Chapter 4 treats the L? spaces of measurable functions f with integrable | f|? for p € (0, 00) and the space L™ of essentially bounded measurable functions on a general measure
- 20. Preface xV space. Here Hélder’s inequality and Minkowski’s inequality are proved for p € (0, oo]. §15 introduces the Banach space and its dual. §16 treats L? spaces for p € [1, oo] as well as for p € (O, 1). §17 treats relation among the L? spaces for different values of p. The £7 spaces of sequences of numbers (a, : n € N) with }7,,cn lan|? < 00 is treated as a particular case of L? spaces in which the underlying measure space is the counting measure space on the set N of natural numbers. The Riesz representation theorem on the L? spaces is proved in §18. §19 treats integration on a locally compact Hausdorff space. Urysohn’s Lemma on the existence of a continuous function with compact support and partition of unity, Borel and Radon measures, the Riesz representation theorem on the space of continuous functions with compact support as well as Lusin’s theorem on approximation of a measurable function by continuous functions are included here. (The placement of §19 in Chapter 4 is somewhat arbitrary.) Chapter 5 treats extension of additive set functions to measures. It starts with extension of an additive set function on an algebra to a measure in §20 and completes the theory with extension of an additive set function on a semialgebra to a measure in §21. (Semialgebra of sets is an abstraction of the aggregate of left-open and right-closed boxes in the Euclidean space R”. Its importance lies in the fact that the Cartesian product of finitely many al- gebras and in particular o-algebras is in general not an algebra, but only a semialgebra.) As an example of extending an additive set function on a semialgebra to a measure, the Lebesgue-Stieltjes measure determined by a real-valued increasing function on R is treated in §22, Theorems establishing the equivalence of the absolute continuity and singularity of a Lebesgue-Stieltjes measure with respect to the Lebesgue measure with the absolute continuity and singularity of the increasing function that determines the Lebesgue-Stieltjes measure are proved. As a second example of extending an additive set function on a semi- algebra to a measure, the product measure on the product of finitely many measure spaces is included in §23. Tonelli’s theorem and Fubini’s theorem on the reduction of a multiple integral to iterated integrals are found here. Chapter 6 specializes in integration in the Lebesgue measure space on R”. In §24 the Lebesgue measure on R” is constructed as an extension of the notion of volumes of boxes in R” to Lebesgue measurable subsets of R”. Then it is shown that the Lebesgue measure space on R” is the completion of the n-fold product of the Lebesgue measure space on R. Regularity of the Lebesgue measure and in particular approximation of Lebesgue measurable sets by open sets leads to approximation of the integral of a measurable function by that of a continuous function. Translation invariance of the Lebesgue measure and integral and linear transformation of the Lebesgue measure and integral are treated. §25 begins with the study of the average function of a locally integrable function. Hardy-Littlewood maximal theorem and Lebesgue differentiation theorem are presented. These are followed by differentiation of a set function with respect to the Lebesgue measure, in particular differentiation of a signed Borel measure with respect to the Lebesgue measure, and density of a Lebesgue measurable set with respect to the Lebesgue measure. §26 treats change of variable of integration by differentiable transformations. Chapter 7 is an introduction to Hausdorff measures on R”. §27 defines s-dimensional Hausdorff measures on R" for s € [0, 00) and the Hausdorff dimension of a subset of R”.
- 21. xvi Preface §28 studies transformations of Hausdorff measures. §29 shows that a Hausdorff measure of integral dimension is a constant multiple of the Lebesgue measure of the same dimension. Every concept is defined precisely and every theorem is presented with a detailed and complete proof. I endeavored to present proofs that are natural and inevitable. Counter- examples are presented to show that certain conditions in the hypothesis of a theorem can not be simply dropped. References to earlier results within the text are made extensively so that the relation among the theorems as well as the line of development of the theory can be traced easily. On these grounds this book is suitable for self-study for anyone who has a good background in advanced calculus. In writing this book I am indebted to the works that I consulted. These are listed in the Bibliography. I made no attempt to give the origin of the theory and the theorems. To be consistent, I make no mention of the improvements that I made on some of the theorems. I take this opportunity to thank all the readers who found errors and suggested improvements in the various versions of the lecture notes on which this book is based. J. Yeh Corona del Mar, California January, 2000
- 22. Preface to the Second Edition In this new edition all chapters have been revised and additional material have been incor- porated although the framework and organization of the book are unchanged. Specifically the following sections have been added: §13 [VI] Length of Rectifiable Curves §15 [VII] Baire Category Theorem [VI] Uniform Boundedness Theorem [IX] Open Mapping Theorem [X] Hahn-Banach Extension Theorems §16 weak convergence in L? spaces in [III] and [TV] of §16 the complete metric spaces L? for p € (0, 1) in [V] of §16 §19 [V] Signed Radon Measures [VI] Dual Space of C(X) §23 [IV.2] Convolution of L? Functions [IV.3] Approximate Identity in Convolution Product [IV.4] Approximate Identity Relative to Pointwise Convergence Besides these topics there are additional theorems in sections: §1, §4, §5, §8, §10, §11, §13, §15, §16, §17, §19, §20, §21, §23, §24, §25, and §27. Also 64 problems have been added. To use this book as a textbook, selection of the following sections for instance makes a possible one-year course at the graduate level: $1 to §13, §15([T] to [VI]), §16 to §21, §23(M] to [III]) It is my pleasure to thank Abel Klein for his helpful comments on the first edition of this book. J. Yeh Corona del Mar, California March, 2006 xvii
- 23. This page intentionally lett blank
- 24. Preface to the Third Edition In this edition several topics are added. Since these topics do not fit in single sections they are presented as appendices. They are: [A] Digital Expansions of Real Numbers [B] Measurability of Limits and Derivatives [C] Lipschitz Condition and Bounded Derivative [D] Uniform Integrability [E] Product-measurability and Factor-measurability [F] Functions of Bounded Oscillation In [B], we show that if the limit of a real-valued function on R exists then it is Borel- measurable. In particular if a real-valued function is differentiable then the derivative is Borel-measurable. In [C], we show that if a real-valued function satisfies a Lipschitz condition on [a, b] C R then it is differentiable a.e. on [a, b] and moreover the derivative is bounded on [a, 5]. In [D], we discuss uniform integrability and equi-integrability. In [F], we define the notion of bounded oscillation for a real-valued function whose domain of definition is a closed box in R”. We show that for the particular case n = 1 a function is of bounded oscillation if and only if it is a function of bounded variation. Also 93 problems have been added. There is now atotal of 394 problems. Corona del Mar, California August, 2013 xix
- 25. This page intentionally lett blank
- 26. List of Notations - ARAAANNZ BX) Bx SBR Bp (Bw), Mt, ‘L amt! a(€) a(€) Miu") Ags Jo, Joes Jeo. Je Yo» Noor Neo» Fe D(f) RS) {D: f <a} Ay Hy (4)" wy R, Mt, u,) R", MT, Hr) the natural numbers the integers the nonnegative integers the real numbers the complex numbers RorC the extended real number system {—co} UR U {oo} the set of ¢ = & + in whereé,n ¢ R the n-dimensional Euclidean space the collection of all subsets of a set X the o-algebra of Borel sets in a topological space X the o-algebra of Borel sets in R the o-algebra of Borel sets in R” the collection of bounded Borel sets in R” the o-algebra of Lebesgue measurable sets in R the o-algebra of Lebesgue measurable sets in R* algebra generated by € o-algebra generated by € o-algebra of .*-measurable sets product o-algebra o (A x B) classes of intervals in R classes of intervals in R" domain of definition of a function f range of a function f abbreviation of {x € D: f(x) < a} Lebesgue outer measure on R Lebesgue measure on R Lebesgue outer measure on R” Lebesgue measure on R” Lebesgue measure space on R Lebesgue measure space on R” R, Mus), Hg) Lebesgue-Stieltjes measure space LP(X, A, 2) L?(X, A, w) £(X, A, uw) p.4 D1.16, p.1 D1.16, p.10 D1.16, p.10 D25.19, p.632 D3.1, p.41 D24.8, p.600 D1.12, p.7 D1.12, p.7 D2.2, p.28 D23.3, p.528 D3.1, p.41 D24.2, p.597 p.19 p.19 p.70 D3.1, p.41 D3.1, p.41 D24.5, p.598 D24.8, p.600 D3.1, p.41 D24.8, p.600 D22.5, p.507 D16.8, p.378 D16.22, p.385 D16.38, p.394
- 27. a L™(X, A, w) Eiog(R" wT, we) L?R, DM, u,) 8(X, A, 4) So(X, A, w) So(X, 2, 2) C(x) ct(X) C.(X) Cy(X) Co(X) c@), CR) c™(R), Co(R) c™@R), CPR) C(R), Co(R) E‘, E°, E, 8E |E| B,(x), B(x, 7) B,(x), Bz, r) 5 (x), SQ, 7) ft, fo RS oF f@+), f@-) supp{f} aT (p; -) Jr(p) at aT lal A<Kp ALp Ie lle I+ llp Il lloo UZ lly.~ Ifill Hy WE H°(E) Fi, F, SE, 98 85, 5°, BS, BS 51 C5 iy K* dim, equal by definition complement, interior, closure, boundary of E diameter of E open ball with center x and radius r closed ball with center x and radius r spherical (hyper)surface with center x and radius r positive and negative parts of a function f teal part of a function f imaginary part of a function f limz ja f(x), limrta f(x) support of f differential of a mapping T at a point p Jacobian matrix of a mapping T at a point p positive and negative parts of a signed measure 1 total variation of a signed measure A absolute continuity of A with respect to jz singularity of 4 and yz uniform norm L? norm essential supremum norm of a bounded linear mapping L of V into W norm of a bounded linear functional f Hausdorff measure HH*(E) as a function ofs € [0, 00) Hausdorff dimension Notations D16.42, p.395 D25.1, p.620 D23.29, p.547 118.2, p.432 L18.2, p.432 T18.3, p.433 D19.48, p.471 D19.48, p.471 D19.9, p.447 D23.45, p.564 123.45, p.564 D23.38, p. 555 D23,38, p. 555 D23.38, p. 555 D23.45, p. 564 D27.1, p.675 D15.33, p.338 D15.33, p.338 D15.33, p.338 D4.25, p.83 p.376 p.376 p.251 D19.7, p.447 p.649 p.650 D10.22, p.213 D10.22, p.213 D114, p.225 D10.16, p.211 T19.50, p.471 D16.8, p.378 D16.36, p.393 D15.29, p.337 D15.39, p344 D27.3, p.675 D27.7, p.677 p.688 127.13, p.680 P27.14, p.681 127.24, p.684 C27.25, p.685 D27.34, p.691
- 28. Notations Euler Fraktur and Script Here is a list of the capital Roman letters, each followed by its corresponding Euler Fraktur and Script letters:
- 29. This page intentionally lett blank
- 30. Chapter 1 Measure Spaces §0 Introduction Let us consider the problem of measuring an arbitrary subset of the real line R. For a finite open interval J = (a, b) in R, we define the length of J by setting £(7) = & — a, and for an infinite open interval J in R, we set £(Z) = 00. Let $3(R) be the collection of all subsets of R. To extend the notion of length to an arbitrary E € 93(R), let (7, : n € N) be an arbitrary sequence of open intervals in R such that ), cy Jn D E, take the sum of the lengths of the intervals >°,cn €Un), and define jz*(E) as the infimum of all such sums, that is, (1) p*(E) = inf { Xnew £m) : (qin €N) such that U,en In D Ej. The set function jz* thus defined on §3(R) is nonnegative extended real-valued, that is, 0 < p*(E) < oo for every E € $3(R), with u*( = 0; monotone in the sense that B*(E) < u*(F) for any E, F € $8(R) such that E C F; and w*(1) = £(D) for every open interval J in R so that j2* is an extension of the notion of length to an arbitrary subset of R. The set function * also has the property that (2) w*(E, U Eo) < w*(E1) + e*(E2) for any two sets F,, Ey € $(R). We call this property the subadditivity of * on $3(R). We say that a set function v on a collection € of subsets of R is additive on € if we have v(E, U £2) = v(E1) + v(E2) whenever £1, Ez € €, Ey 1 Ey = Gand E, U Ep € €. Our set function j4* is not additive on $3(R), that is, there exist subsets FE, and Ey of R which are disjoint but not separated enough, as far as z* is concerned, to have u*(E, U E2) = B*(E1) + w*(E2). Examples of such sets are constructed in §3 and §4. Let us show that it is possible to restrict 4* to a subcollection of $8(R) so that p* is additive on the subcollection. Let Z € 9(R) be arbitrarily chosen. Then for every A€ (BR), ANE and AN E*, where E° is the complement of E, are two disjoint members of $8(R) whose union is A. We say that the set EF satisfies the .*-measurability condition and E is a jz*-measurable set if é)) ut(A) = p*(ANE)+ eu (ANE*) forevery A € PR). 1
- 31. 2 CHAPTER 1 Measure Spaces Itis clear that if E satisfies condition (3), then so does E°. Note also that J and R are two ex- amples of members of $3(R) satisfying condition (3). Now let )t(*) be the subcollection of §3(R) consisting of all 2*-measurable sets in 93(R). Let us show that 2t(*) is closed under unions. Let £1, Ez € Dt(u*). Then we have 4*(A) = 2*(AN Ey) + u*(AN Ef) for an arbitrary A € $$(R) by (3). With AN Ef as atesting set for our E2 € Nt(u*) replacing A in condition (3), we have u*(A M Ef) = p*(AN Ej N Eo) + w*(AN Ef N E5). Thus B*(A) = w*(AN Ey) + p*(AN EN Ey) + w(AN EL N ES). For the union of the first two sets on the right side of the last equality, we have (ANE1)U(AN(E{NE2)) = AN(E,UCE{NE2)) = AN(E1U(E2 Ey) = AN(E1UE2). Then *(AN E1) + 2*(AN (Ef E2)) > u*(AN (Ei U Ea)) by the subadditivity of p* on §3(R). Thus we have 2*(A) > w*(AN (EU Ex)) + u*(AN (£1 U E2)*). On the other hand by the subadditivity of 2* on $3 (R), the reverse of this inequality holds. Thus £1 U £2 satisfies condition (3) and is therefore a member of 9J%(*). This shows that Nt(u*) is closed under unions. We show in §2 that 29t(*) is closed under countable unions. A collection of subsets of a nonempty set X is called a o-algebra of subsets of X if it includes X as a member, is closed under complementations and countable unions. Thus our St(jz*) is ac-algebra of subsets of R. Let us show next that jz* is additive on the o-algebra Nt(z*) of subsets of R. Thus let £,, £2 € Dt(z*) and assume that Ej NM Ey = §. Now with E, U E, as the testing set A in the z*-measurability condition (3) which is satisfied by £1, we have *(E1 U Bo) = w*((E1 U Ex) 9 E1) + w*((E1 U E2) 9 Ef). But (EZ; U Eo) Ey = Ey and (Ey U Ey) N Ef = Eo. Thus the last equality reduces to (4) #* (Ey U Ea) = w*(E1) + w* (Ep). This shows that jz*, though not additive on $3(R), is additive on the subcollection Nt(u*) of PR). Now pz* is additive on MNt(*) so that we may regard it as the extension of the notion of length to sets which are members of Nt(z*). For this extension j:* to be interesting, the collection )t(j.*) must be large enough to include subsets of R that occur regularly in analysis. In §3, we show that 9)t(2*) includes all open sets in R and all subsets of R that are the results of a sequence of such set theoretic operations as union, intersection, and complementation, on the open sets.
- 32. §1 Measure on a o-algebra of Sets 3 §1 Measure on a c-algebra of Sets [I] o-algebra of Sets Notations. We write N for both the sequence (1, 2, 3, .. .) andthe set {1, 2, 3, ...}. Whether a sequenice or a set is meant by N should be clear from the context. Similarly we write Z for both (0, 1, —1, 2, —2, ...) and {0, 1, -1, 2, —2, ...} and Z, for both (0, 1, 2,...) and {0, 1,2,...}. Definition 1.1. Let X be an arbitrary set. A collection A of subsets of X is called an algebra (or a field) of subsets of X if it satisfies the following conditions: 1° XeQ, 2 AEM=> AEA, 3 A,BEA>AUBEA. Lemma 1.2. If 2 is an algebra of subsets of a set X, then (1) GEM, (2) At,..., An € A= iy Ae € A, G3) A BeADBANBEA, (4) At,..., An € A= (fy Ac € A, 6) A BeA>ABem. Proof. (1) follows from 1° and 2° of Definition 1.1. (2) is by repeated application of 3°. Since AN B = (A‘ U B°)*, (3) follows from 2° and 3°. (4) is by repeated application of (3). For (5) note that A B = AN BS € BW by 2° and (3). wf Definition 1.3. An algebra A of subsets of a set X is called a o-algebra (or a o-field) if it satisfies the additional condition: 4° (An:n EN) CA Une An € A Note that applying condition 4° to the sequence (A, B, 0, 8, ...), we obtain condition 3° in Definition 1.1, Thus 3° is implied by 4°. Observe also that if an algebra 2 is a finite collection, then itis ao-algebra. This follows from the fact that when 2 is a finite collection then a countable union of members of 2 is actually a finite union of members of 2 and this finite union is a member of & by (2) of Lemma 1.2. Lemma 1.4. If & is a o-algebra of subsets of a set X, then (0) (An:n EN CAD (ey An € A Proof. Note that (en 4n = (Upen AS)°- By 2°, A$ € and by 4°, nen Ag € Thus by 2°, we have (U,ey AS)" € 2.
- 33. 4 CHAPTER 1 Measure Spaces Notations. For an arbitrary set X, let $3(X) be the collection of all subsets of X. Thus A & §B(X) is equivalent to A C X. Example 1. For an arbitrary set X, ¥3(X) satisfies conditions 1° - 3° of Definition 1.1 and condition 4° of Definition 1.3 and therefore it is a o-algebra of subsets of X. It is the greatest a-algebra of subsets of X in the sense that if 2l is a o-algebra of subsets of X and if P(X) C A thenA = F(X). Example 2. For an arbitrary set X, {4, X} is a o-algebra of subsets of X. It is the smallest o-algebra of subsets of X in the sense that if 2% is a o-algebra of subsets of X and if Ac {B, X} then A = {B, X}. Example 3. In R?, let 9% be the collection of all rectangles of the type (a1, bi] x (a2, bz] where —oo < a; < b; < oo fori = 1, 2 with the understanding that (a;, 00] = (a, 00). Let & be the collection of all finite unions of members of $t. We have Ht C A since every A € 9 is the union of finitely many, actually one, members of 9% so that A € . We regard § as the union of 0 members of 9% so that @ € &. It is easily verified that Mis an algebra of subsets of R?. However & is not a o-algebra. Consider for instance, An = (n—4,n] x ©, 1] € CM forn EN. Then U),ey An is not a finite union of members of 9 and is thus not a member of 2. [II] Limits of Sequences of Sets Definition 1.5. Let (A, : n € N) be a sequence of subsets of a set X. We say that (A, : n € N) is an increasing sequence and write An t if An C Anti forn € N. We say that (A, : n € N) is a decreasing sequence and write A, | if An D> Anyi forn ENA sequence (A, : n € N) is called a monotone sequence if it is either an increasing sequence or a decreasing sequence. For an increasing sequence (A, :n € N), we define qd) lim An = U» = {x € X:x € A, for somen € N}. RE. For a decreasing sequence (A, : n € N), we define (2) bm An = f) An = {x €X:x € A, foreveryne N}. neN For a monotone sequence (A, :n € N), im, A, always exists although it may be 9. If A, ¢, then jim An = @if and only if A, = 9 for every n € N. If A, |, we may have tm, An = @ even if A, # for every n € N. Consider for example X = R and An = (0,2) forn € N. Then lim A, =. On the other hand if A, = [0, }) forn e N n00 then A, | and jim, A, = {0}. In order to define a limit for an arbitrary sequence (A, : n € N) of subsets of a set X we define first the limit inferior and the limit superior of a sequence.
- 34. §1 Measure on a o-algebra of Sets 5 Definition 1.6. We define the limit inferior and the limit superior of a sequence (A, : n € N) of subsets of a set X by setting ) liminfA, = J (7) 4x, neNk>n (2) lim supAy = =U 4. neNkzn Note that (M,>n Ax : 2 € N) is an increasing sequence of subsets of X and this implies that lim Msn At = Unew Mien Ae exists. Similarly (Usen At: 2 €N) is a decreas- ing sequence of subsets of X and thus lim, Ubon Se = Onew Upon Ax exists. Thus lim inf A, and lim sup A,, always exist although they may be 9. noo no Lemma 1.7. Let (A, : n € N) be a sequence of subsets of a set X. Then (1) liminf 4, = {x € X: x © A, for all but finitely many n € N}. Q) lim sup A, = {x € X: x € A,for infinitely many n € N}, @) liming A, clim sup Ap. n—>00 Proof. 1. Letx ¢ X. If x € A, for all but finitely many n ¢ N,then there exists no ¢ N such thatx € A; for all k > ap. Then x € (gong At C nen Mion Ak = liminf An. Conversely if x € lim inf An = Unen Mean Ate then x € xn, Ax for some np € N and * 2 2 thus x € A, for all k > no, that is, x € A,, for all but finitely many n € N. This proves (1). 2. Ifx € A, for infinitely many n ¢ N, then for everyn ¢ N we havex € (Jj,, Ax and thus x € Mew Upon At = lim sup Ay. Conversely ifx € limsup An = (ye~ Uson Abs n>00 100 then x € Uben Ae for every n € N. Thus for every n € N,x € Ax for some k > n. This shows that x € A, for infinitely many n € N. This proves (2). 3. (1) and (2) imply (3). & Definition 1.8. Let (A, : n € N) be an arbitrary sequence of subsets of a set X. If lim inf A, = jim sup Ap, then we say that the sequence converges and define im, An by noo > OO setting jim, Ae= lim inf An = = lim sup A,,. if liminf A, % limsup A, then. ‘tim, An n00 n> does not exist Note that this definition of lim A, contains the definition of lim A, for monotone n00 n—>00 sequences in Definition 1.5 as particular cases and thus the two definitions are consistent. Indeed if A, ft then (>, 4¢ = An for every n € N and U,en( ion At = Unen An and therefore liminf An = Unen An. On the other hand, L),, At = Uxen At for every n € Nand (yew Upon At= Uber Ax and thus lim sup An = Unen An- Similarly for An J. Note also that if (A, : n € N) is such that liminf Ay = = Q and lim sup A, = @ also n00
- 35. 6 CHAPTER 1 Measure Spaces then lim A, = @. n-00 Example. Let X = R and let a sequence (A, : n € N) of subsets of R be defined by Ai = [0,1], 43 = [0,3], 4s = [0, 5],-.- and Ao = [0, 21, Ay = [0,4], Ag = [0, 6], .... Then lim inf An = {x € X: x € A,for all but finitely many n € N} = {0} and lim sup A, = {x € X : x € A, for infinitely many n € N} = [0, 00). Thus jim, An does ny notexist, The subsequence (Ap, : k € N) = (Aj, A3, As, .. .) is adecreasing sequence with jimn An, = {0} and the subsequence (A,, : k € N) = (Az, Aq, Ag, ...) is an increasing sequence with jim n An = [0, 00). Theorem 1.9, Let 2 be a o-algebra of subsets of a set X. For every sequence (A, :n € N) in &, the two setslimminf A, and lim supAy,, are in 2. So is im a An ifit exists. noo Proof. For every n € N, (yo, Az € A by Lemma 1.4. Then nen Mon At € A by 4° of Definition 1.3. This shows that lim inf An € &. Similarly U,., Ak € A by 4° of Definition 1.3. Then cn jon At "2eA. by Lemma 1.4. Thus limsup A, € &. If 00 lim A, exists, then lim A, = liminf A, € &. . n—>0o noo noo [III] Generation of o-algebras Let A be an arbitrary set. If we select a set E, corresponding to each a ¢€ A, then we call {E, : a € A} a collection of sets indexed by A. Usual examples of indexing set A are for instance N = {1, 2,3,...}, Z= (0, 1, —1,2, —2,...}, and Z, = {0,1,2,...}. An arbitrary set A can serve as an indexing set. Lemma 1.10. Let {2l, : a € A} be a collection of o-algebras of subsets of a set X where A is an arbitrary indexing set. Then 1yc4 Ua is a a-algebra of subsets of X. Similarly if {2y : @ € A} is an arbitrary collection of algebras of subsets of X, then yc4 Aa is an algebra of subsets of X. Proof. Let {M. : @ € A} be an arbitrary collection of o-algebras of subsets of X. Then Nee A, is a collection of subsets of X. To show that it is a o-algebra we verify 1°, 2°, and 3° in Definition 1.1 and 4° in Definition 1.3. Now X € 2%, for every a € A so that X € (ea Me verifying 1°. To verify 2°, note that if E € (,c4 Me, then E € A, so that E° € Ay for every a € A and then E° € (,-4 Ay. 3° is implied by 4°. To verify 4°, let (E, > n EN) C Oye4 Me. Then for every a € A, we have (E, : n € N) C Ay so that Unen En € My. Then Unen E,€ Mees Mo. Ot Theorem 1.11. Let € be an arbitrary collection of subsets of a set X. There exists the smallest o-algebra Ag of subsets of X containing €, smallest in the sense that if U is a o-algebra of subsets of X containing € then Ay C A. Similarly there exists the smallest algebra of subsets of X containing €.
- 36. §1 Measure on a o-algebra of Sets 7 Proof. There exists at least one o-algebra of subsets of X containing €, namely $3(X). Let (Aq : a € A} be the collection of all o-algebras of subsets of X containing €. Then Nacsa Mo Contains € and it is ao-algebra according to Lemma 1.10. Itis indeed the smallest o-algebra containing € since any a-algebra &% containing € is a member of {2, : a € A} so that AD ge, Ma. Definition 1.12. For an arbitrary collection € of subsets of a set X, we write o (€) for the smallest o-algebra of subsets of X containing € and call it the o-algebra generated by €. Similarly we write a(€) for the smallest algebra of subsets of X containing € and call it the algebra generated by €. It follows immediately from the definition above that if €; and €2 are two collections of subsets of a set X and €; C €o, then o(€1) C o(€z). If A is a o-algebra of subsets of X, then o (2) = 2. In particular for an arbitrary collection € of subsets of X, we have a(o(€)) =a(€). Let f be a mapping of a set X into a set Y, that is, f is a Y-valued function defined on X. The image of X by f, f(X), is asubset of Y. Let E be an arbitrary subset of Y. E need not be a subset of f (X) and indeed E may be disjoint from f(X). The preimage of E under the mapping f is a subset of X defined by f—!(E) = {x € X : f(x) € E}, that is, the collection of everyx € X such that f(x) € E. Thus if EM f(X) = @ then fo) = 6. For an arbitrary subset E of Y we have f(f—!(E)) C E. Note also that f7W) =x, STE) = $70 BE) = FOO) FI) = X FI = (Fy, I7"(Uaes Ea) = Uaes f-"(Ex)s I" (Maes Bu) = Ques f-"(Ea). For an arbitrary collection € of subsets of Y, let f7(€) = {f(E) :EeE€}. Proposition 1.13. Let f be a mapping of a set X intoa set Y. If B is ao-algebra of subsets of Y then f—'(98) is a o-algebra of subsets of X. Proof. Let us show that 2% is a c-algebra of subsets of X by showing that X € f—1(98), f71(93) is closed under complementations in X, and f —1(98) is closed under countable unions. 1. We have X = f—!(¥) € f—1(98) since ¥ € B. 2. Let A € f—!(93). Then A = f—!(B) for some B € 98. Since B° € %B, we have f7-(B°) © f-1(3). On the other hand, f—1(B°) = (f-1(B))* = A®. Thus we have Ae € f-1(8). 3. Let (A, : 2 € N) be an arbitrary sequence in f 7108). Then A, = f (Bn) for
- 37. 8 CHAPTER 1 Measure Spaces some B, € % for each n € N. Thus we have Un =U #72) = FU Be) € £1) neN neN neN since |_),<n Bn € 3. This verifies that f 16598) isa o-algebra of subsets of X. a Theorem 1.14. Let f be a mapping of a set X into a set Y. Then for an arbitrary collection €of subsets of Y, we have a(f-() =f (o(€)). Proof. Since € C o(€), we have f-1(€) c f-!(o(€)) and consequently o (f-1(€)) c a(f—1(o(€))). Since o(€) is a o-algebra of subsets of Y, f—!(a(€)) is a c-algebra of subsets of X by Proposition 1.13 so that o(f—!(o(€))) = f—'(o(€)). Thus we have a(f-"@) c fo). To prove the reverse inclusion, let 2%, be an arbitrary o-algebra of subsets of X and let MW ={AcY: fA) eA}. To show that 22 is a o-algebra of subsets of Y, note first ofall that f ly) = X € A sothat Y € M>. Secondly, for every A € Wo we have f—1(A*) = (f-1(A))° € Bi so that AC € >. Finally for any (A, :n € N) C Mo, we have f-! (ney An) = Unen £1 (An) € Mi so that L,en 4n € Mo and thus 2%, is a o-algebra of subsets of Y. In particular, if we let a={AcY: fA) eo(f))}, then % is a o-algebra of subsets of Y. Clearly & > € and thus 2% D> o(€) and then f7Q) Dd f-(a(€). But f(D) c o(f—1(€)) by the definition of 2. Thus we have o(f-(€)) > f-1(o(€)). Therefore o (f—1(€)) = f-t(o(@)). Notations. For an arbitrary collection € of subsets of a set X and an arbitrary subset A of X, letus write ENA ={ENA: E € €}. We write o4(€ 2 A) for the o-algebra of subsets of A generated by the collection € N A of subsets of A. Note that the subscript A in og indicates that it is a o-algebra of subsets of A, not a o-algebra of subsets of X. Theorem 1.15. Let € be an arbitrary collection of subsets of a set X andlet A C X. Then a4(EN A) =G(E)NA. Proof. Since € C o(€) wehave ENA C o(€)NA. From the fact that o (€) is ao-algebra of subsets of X and A C Xit follows that o(€) N A is a o-algebra of subsets of A. Thus () os(EN A) Co(E)NA. Therefore, to prove the theorem it remains to show @) a(€)NA Coa(EN A).
- 38. §1 Measure on a o-algebra of Sets 9 Let & be the collection of subsets K of X of the type (3) K =(CNA‘)UB, where C € o(€) and B € o4(EM A). Observe that since B C A, the union in (3) is a disjoint union. By (3), X € # and K is closed under countable unions. To show that & is also closed under complementations, let K € & be as given by (3). Then K°=XK =[(KNA)UA][(CNA)UB] =[(XN AD) (CN AY] U(A B) since XN A° > CN AC and A > B. But (XM A°) (CN.A*) = C*N A*. Therefore K° =(C°NA)U(A BER. Thus & is closed under complementations and is therefore a o-algebra of subsets of X. Next, observe that for any K € as given by (3) we have KM A = B € og(EN A) 80 that RN A C o4(EM A). Thus to show (2) it suffices to show that o(E)N AC ANA. Since & is a o-algebra of subsets of X, it remains only to show that€ Cc A. LetEe € and write FE = (EN A°) U(EN A). Since EN A € o4(EN A), E is a subset of X of type (3). Thus E € & and therefore € Cc &. This completes the proof. [TV] Borel o-algebras To fix our terminology let us review definitions of some topological concepts. Let X be a set. A collection of subsets of X is called a topology on Xif it satisfies the following axioms: I ged, 0 Xe, I {Eyg:a€ A}CO> Ube ED, IV Ei,E,ED > E()EeD. The pair (X, 9) is called a topological space. The members of 9 are called the open sets of the topological space. A subset E of X is called a closed set if its complement E° is an open set. Thus X is both an open set and a closed and so is 9. An arbitrary union of open sets is an open set and a finite intersection of open sets is an open set. An arbitrary intersection of closed sets is a closed set and a finite union of closed sets is a closed set. The interior E° of a subset E of X is defined as the union of all open sets contained in E. Thus it is the greatest open set contained in E. The closure E of E is defined as the intersection of all closed sets containing E. It is the smallest closed set containing E. The boundary 0£ of £ is defined by 2E = (E° Uu (E))". A subset £ of X is called a compact set if for every collection 23 of open sets such that EC Uves V there exists a finite subcollection {Vj, ..., Vw} such that EC Ura Va.
- 39. 10 CHAPTER 1 Measure Spaces Let X be an arbitrary set. A function p on X x X is called a metric on X if it satisfies the following conditions: 1° p(@, y) € [0, 00) for x, y € X, 2 pG,y) =0x=y, 3° p@, y) = py, x) forx,y € X, 4° triangle inequality: p(x, y) < p(x, z) + p@, y) forx, y,z€ X. The pair (X, p) is called a metric space. In R®, if we define p(x, y) = lx — yl = {Df Ge — ye)2}'” for x = Gr, an) and y = (y1,..., yn) in R’, then ¢ satisfies conditions 1°, 2°, 3°, and 4° above and is thus amettic. This metric on R” is called the Euclidean metric. Ina metric space (X, p), if xo € X andr > Othe set B(xo, r) = {x € X : p(x, x0) <r} is called an open ball with center xo and radius r. A subset E of X is called an open set if for each x € E there exists r > 0 such that B(x, r) C E. An open ball is indeed an open set in the sense defined above. The collection of all open sets in a metric space satisfies axioms I, I, II, and IV and is thus a topology. We call this topology the metric topology of X by the metric p. A set E in a metric space (X, ) is said to be boundedif there exist x9 € X andr > 0 such that E C B(xo,r). A set E in R” is a compact set if and only if E is a bounded and closed set. Definition 1.16. Let D be the collection of all open sets in a topological space X. We call the o-algebra o (2) the Borel o-algebra of subsets of the topological space X and we write 38x or B(X) for it. We call its members the Borel sets of the topological space. Lemma 1.17. Let € be the collection of all closed sets in a topological space (X,). Then o(€) =o0(D). Proof. Let E €¢ €. Then E° € O C a(Q). Now since o(D) is a o-algebra, we have E = (E°Y € o(). Thus € C o(D) and consequently o(€) c a (o()) = o(). By the same sort of argument as above we have o(D) C o(€). Therefore o(€) =o (D). © Definition 1.18. Let (X, 9) be a topological space. A subset E of X is called a G-set if it is the intersection of countably many open sets. A subset E of X is called an F,-set if it is the union of countably many closed sets. Thus, if EF is a G-set, then E° is an F,-set, and if E is an F,-set then E° is a Gs-set. Note that every G5-set is a member of Sy. So is every F,-set. Indeed if E is a G;-set, thenE = (),cn On where O, € O forn € N. Now O, € 0 C o(D) = Bx for every nN. Since %3y is a o-algebra, we have E = (),cy On € Bx. Let us note also that if Z is a G3-set, then there exists a sequence (O, : n € N) of open sets such that E = (),cy On. If we let Gz = (eu1 Ox, then (G, :n € N) isa decreasing sequence of open sets and cy Ga = Mew On = E. Thus a Gs-set is always the limit of a decreasing sequence of open sets. Similarly if E is an F,-set, then there exists
- 40. §1 Measure on a o-algebra of Sets 11 a sequence (C,, : n € N) of closed sets such that E = nw Cy. If we let F, = Ula Ck, then (F, : n € N) is an increasing sequence of closed sets and nen Fn = Une Cn = E- Thus an F,-set is always the limit of an increasing sequence of closed sets. [V] Measure on a c-algebra Notations. Let R = {—oo} URU {co} and call it the extended real number system. We use the alternate notation [—oo, co] for R also. Definition 1.19. Let € be a collection of subsets of aset X. Let y be anonnegative extended real-valued set function on €. We say that (a) y is monotone on € if y(E1) < y(E2) for Ei, Ez € € such that E, C E2, (b) y is additive on € if y(E1 U E2) = y(E1) + y(E2) for E1, Ez € € such that E, NE. =@and E, UE. € €, (© y is finitely additive on € if y (U1 Ex) = Di y (En for every disjoint finite sequence (Ex: k =1,...,n) in € such that Ufa E,e €, (d) y is countably additive on € if vy (Uncx En) = Vnen ¥ (En) for every disjoint sequence (Ey, :n € N) in € such that |), cy En € € (©) y is subadditive on € if y(E, U Eo) < y(E1) + y(E2) for E1, Eo € € such that E,UE€& (@ y is finitely subadditive on € if y (Uha Ex) < Di (Ex) for every finite sequence (Ex :k =1,...,n) in € such that (Z_; Ex € €, (g) y is countably subadditive on € if y (Unen En) < Yyen Y (En) for every sequence (En in €N) in € such that ncn En € €. Note that in (c) while LJf_, Ex € € is required, it is not required that any of J2_1 Ex. Ubi Et, ..., U2zi Ex be in €. Note also that (c) implies (b) and (f) implies (¢). Observation 1.20. Let y be a nonnegative extended real-valued set function on a collection € of subsets ofa set X. Assume that # € € and y(@) = 0. (a) If y is countably additive on €&, then it is finitely additive on €. (b) If y is countably subadditive on €, then it is finitely subadditive on €. Proof. Suppose y is countably additive on €. To show that it is finitely additive on €, let (Ex : k = 1,...,m) be a disjoint finite sequence in € such that Ute Eye €. Consider the infinite sequence (F, : k € N) in € defined by & = E, fork = 1,...,n and F, = fork >n+1. Since# € €, (RH : k € N) is a disjoint sequence in € with Uben Fe = Uf Ex € €. Thus by the countable additivity of y on € and by the fact that y(B) = 0, we have y (Ut Et) = » (Uren Fe) = Daew (Fe) = Cs y (Ex). This proves the finite additivity of y on €. We show similarly that if y is countably subadditive on &, then it is finitely subadditive on €. Lemma 1.21, Let (£,, : n € N) be an arbitrary sequence in an algebra X of subsets of a set X. Then there exists a disjoint sequence (F, : n € N) in Q such that
- 41. 12 CHAPTER 1 Measure Spaces N N (1) Ua=U" for every N EN, a=1 n=1 and ® Un=U-A. neN neN In particular, if & is a o-algebra, then nen Fa = Unen En € &. Proof. Let F, = E, and F, = E, (E, U...U E,_1) forn > 2. Since 2 is an algebra, F, € A forn ¢ N. Let us prove (1) and (2) and then the disjointness of (F, : n € N). Let us prove (1) by induction. To start with, (1) is valid when N = 1 since Fy = Ej. Next, assume that (1) is valid for some N € N, that is, Usa E, = Ux, F,,. Then we have N+1 N N N N+ U me =(U%) U Frat = (U En) U (Eve U 2a) = U Ee n=l a=1 n=l n=l n=l that is, (1) holds for N + 1. Thus by induction, (1) holds for every N € N. To prove (2), let x € Unen E,. Then x € E,, for some n € N and thus we have % © ger Ex = Ufa Fe C Ujen Fn by (1). We show similarly that ifx € Ucn 7, then x € nen En. Thus we have en En = Unen Fn. This proves (2). Finally let us show that (F, : n € N) is a disjoint sequence. Consider F,, and F, where n #m, sayn <m. We have Fy, = Em (FE, U---U Em_1). By (1) and by the fact that n <m, wehave £1 U---U En-1 = Fi U---U Fy-1 D Fy. Thus we have Fy) Fn = @. This prove the disjointness of (F,:néN). Lemma 1.22. Let y be a nonnegative extended real-valued set function on an algebra A of subsets of a set X. (a) If y is additive on &, it is finitely additive, monotone, and finitely subadditive on A. (b) If y is countably additive on U, then it is countably subadditive on A. Proof. 1. Suppose y is additive on A. Let (Ej : k = 1, ..., m) bea disjoint finite sequence in 2. Since M is an algebra, we have UE, E; € Afork = 1,...,n. By the disjointness of Ut E; and E,, and by the additivity of y on 2, we have (Ue) = AU Ex) + (En). k=! k=1 Repeating the argument, we have y (Uj_1 Ex) = Df, y(Ex). This proves the finite additivity of y on 2. To prove the monotonicity of y on A, let £1, Eo € Wand Ey C Eo. Then £1, Eo E, € A, Ey NCE. £1) =G, and Ej U (£2 £1) = Eo € Aso that by the additivity of y on &, we have y(E1) + y(E2 £1) = y (£2). Then since y (#2 F1) = 0, we have y(EZ1) < y(£2). This proves the monotonicity of y on 2. To show the finite subadditivity of y on 2, let (EZ; :k = 1,...,n) be a finite sequence in A. If we let Fj = E, andFH = E; (E, U---U Ex_1) fork = 2,...,m, then as we showed in the Proof of Lemma 1.21, (Fi, : k = 1,...,m) is a disjoint finite sequence in
- 42. §1 Measure on a o-algebra of Sets 13 A with Ut_, Fe = Ufir Ex so that by the finite additivity and the monotonicity of y on MA, we have y (Up_y Ex) = v (Utar Fe) = D1 v Fe) < D1 v (Ex). This proves the finite subadditivity of y on 2f. 2. Suppose y is countably additive on 2{. To show that it is countable subadditive on A, let (E, : n € N) be a sequence in & such that ),.y En € UW. Let FP) = Ey and Fy = E, (2. VU... E,-1) forn > 2. Then by Lemma 1.21, (F, : n € N) is a disjoint sequence in & and en Fn = Unen En. Thus by the countable additivity and the monotonicity of y on & by (a), we have ¥ (new En) = ¥ (nen Fn) = Lnen (Fn) = Donen ¥ (En). This proves the countable subadditivity of y on 2. Proposition 1.23. Let y be a nonnegative extended real-valued set function on an algebra A of subsets of a set X. If y is additive and countably subadditive on A then y is countably additive on A. Proof. Suppose y is additive and countably subadditive on 2. To show that y is countably additive on 2, let (E, : n € N) be a disjoint sequence in %& such that |),<y En € 2. The additivity of y on 2% implies its monotonicity and finite additivity on 2by (a) of Lemma 1.22. Thus for every N ¢ N, wehave y(Unen En) = ¥(US, En) = 01 v(En). Since this holds for every N € N, we have y(Unen En) = Donen ¥(En)- On the other hand, by the countable subadditivity of y on &, we have y(U,ew En) < Lnen (En). Thus v(Unen Ex) = Ynen (En). This proves the countable additivity of y on 2. Definition 1.24. Let A be a o-algebra of subsets of a set X. A set function p defined on A is called a measure if it satisfies the following conditions: 1° nonnegative extended real-valued: p(E) € [0, 00] for every E € &, 2° 4) = 0, 3° countable additivity: (En :n © N) CA, disjoint => u (Open En) = Cen H(En)- Lemma 1.25. A measure u on a o-algebra A of subsets of a set X has the following properties: (1) finite additivity: (E,, ..., E,) C A, disjoint > w (Uta Ex) = Dee MED, (2) monotonicity: E,, Ez € A, Ey C FE. > w(E1) < w(E2), G) Ei, By € &, Ey C Ey, w(E1) < 00 = w(E2 Ei) = w(E2) — w(ED, (4) countable subadditivity: (En :n € N) CM => w (hen En) < nen (En) and in particular (5) finite subadditivity: (E1,..., En) CM => w (Uta Ex) < DL w(Ed- Proof. The countable additivity of ~ on A implies its finite additivity on 2% by (a) of Observation 1.20. The finite additivity of jz on 2% implies its additivity on 2{ and then its monotonicity on 2 by (a) of Lemma 1.22. To prove (3), let £1, Hz € Wand Ey C Ey. Then £; and E> E; are two disjoint
- 43. 14 CHAPTER 1 Measure Spaces members of 2 whose union is equal to E27. Thus by the additivity of 2 on 2, we have B(E2) = B(E}) + e(E2 £1). If w(E1) < 00, then subtracting (£1) for both sides of the last equality, we have (£2) — (£1) = (E2 £1). This proves (3). The countable additivity of jz on 2 implies its countable subadditivity on & by (b) of Lemma 1.22. This then implies the finite subadditivity of 4. on 21 by (b) of Observation 1.20. & Regarding (3) of Lemma 1.25, let us note that if 4(£,) = oo then by the monotonicity of 2 we have (Ez) = 00 also so that 4.(£2) — w(E1) is not defined. [VI] Measures of a Sequence of Sets Let be a measure on a a-algebra & of subsets of a set X. Let (EZ, : n € N) be a sequence in 2. If lim E&, exists, does lim y(E,) exist? If it does, do we have n> noo uf jim, En) = jim, uU(E,)? The next theorem addresses this question for monotone se- quences of measurable sets. It is based on the countable additivity of a measure. It is a fundamental theorem in that a subsequent theorem regarding the limit inferior and the limit superior of the measures of an arbitrary sequence of measurable sets as well as the monotone convergence theorem for the Lebesgue integral, Fatou’s lemma, and Lebesgue’s dominated convergence theorem are ultimately based on this theorem. Theorem 1.26. (Monotone Convergence Theorem for Sequences of Measurable Sets) Let be a measure on a o-algebra A of subsets of a set X and let (E, :n € N) bea monotone sequence in Q. (a) If En t, then lim j(En) = u4(lim Ey). no 00 (b) FE, J, then lim p(E,) = a iim Ex), provided that there exists a set A € UA with n—>00 > OO HAA) < 00 such that Ey Cc A. Proof. If Z,, +, then dim, En = Unen En € Wl. If Ey J, then dim, En = (hen En € A. Note also that if (E, : n € N) is a monotone sequence in 2, then (u(Z,) : 2 € N) isa monotone sequence in [0, oo] by the monotonicity of jz so that jim (En) exists in [0, oo]. 1. Suppose E,, +. Then we have u(Z,) +. Consider first the case where 2(Ey,) = 00 for some mo € N. In this case we have Aim, w(En) = oo. Since Eno C Unen En = lim E,, we have jt( lim E,) > (Eno) = 00. Thus j4( lim E,) =0co= lim (Ep). noo noo ROO n> 00 Consider next the case where j4(E,) < 00 for every n € N. Let Ey = G and consider a disjoint sequence (F, : n € N) in 2 defined by F,, = E, En-1 forn € N. We have En = UN, Fa for every N € N and hence Unen En = Unen /n- Then we have u( Jim En) = 2((JEn) = (Um) = a) neN neN neN =oEn Env) =O {wEn) — o(En-v)}, neN ncN
- 44. §1 Measure on a o-algebra of Sets 15 where the third equality is by the countable additivity of yz. and the fifth equality is by (3) of Lemma 1.25. Since the sum ofa series is the limit of the sequence of partial sums we have Do {eGin) — e(En—1)} = lim "Y7 {a Ex) — uEx-1)} k=1 neN = Jim {w(En) — w(Eo)} = lim (En). Thus we have j2( lim E,) = lim (E,). AO n> 0O. 2. Suppose E, | and assume the existence of a containing set A with finite measure. Define a disjoint sequence (F, : n € N) in 2 by setting F, = Ey En41 form € N. Then a A(h=Ur,. neN neN To show this, letx € £1 Myen En. Thenx € EF, and x is not in every E,. Since Ey |, there exists the first set E,.+41 in the sequence not containing x. Then x € Eng Engt+i = Fag C Unen Fn. This shows that Ey Men En C Unen Fn. Conversely ifx € Unen Fas thenx € Fry = Eng Eno+1 for some no € N. Now x € En, C Ey. Sincex ¢ Engii, We have x ¢ pen En. Thus x € E; Mex En. This shows that Len Fn C Ei Open En Therefore (1) holds. Now by (1), we have @) w(E1 7) En) =H( UF): neN neN Since 4 (Qnen En) < #(E1) < (A) < 00, we have by (3) of Lemma 1.25 @) ——w(Z1 7) Ba) =n) — 2( 1) Bn) = wy) — wfJim 2). neN neN By the countable additivity of 4, we have @) u( U Fa) = 0 wa) = 7 wa Ens) neN neN nen =) {oGn) — wEny1)} = fim YY fa (Ee) — a Ber)} neN k=1 = im {w(E1) — w(En41)} = aE) — fim, u(En+1)- Substituting (3) and (4) in (2), we have H(E1) — wf Jim En) = 2(Ei) — lim (En41) = #(E1) — lim jo(Ep). Subtracting (£1) € R from both sides we have a lim, En) = jim, (En). O
- 45. 16 CHAPTER 1 Measure Spaces Remark 1.27. (b) of Theorem 1.26 has the following particular cases. Let (FE, : n € N) be a decteasing sequence in %. Then lim (En) = a lim, E,) if any one of the following conditions is satisfied: (a) U(X) < 00, (b) u(E1) < «0, (c) 4(En,) < 00 for some no € N. Proof. (a) and (b) are particular cases of (b) of Theorem 1.26 in which X and FE) respectively are the containing set A € 21 with 4(A) < oo. To prove (c), suppose 4(En,) < 00 for some no € N. Let (F;, : 2 € N) be a decreasing sequence in 21 obtained by dropping the first no terms from (EZ, : n € N), that is, we set F,, = Engin forn € N. Lemma 1.7 implies that lim inf i= lim inf E,, and lim sup F, = n>00 lim sup EZ, and thus lim F, = lim E,. Now since (F, : n € N) is a decreasing sequence noo noo n->00 and F, C En, forn € N and since 4(En,) < 00, (b) of Theorem 1.26 applies so that lim »(F,) = u({ lim F,) = ( lim E,). Since (u(F,) : n € N) is a sequence obtained noo noo noo by dropping the first ng terms of (%(E,) : n € N), we have im, u(F,) = im, H(E,). Therefore we have lim y(E,) = 2{ lim Ep). a) n00 Let yz be a measure on a o-algebra A of subsets of a set X. Then for an arbitrary sequence (E, :n € N) in 2, lim inf E,, and lim sup E,, exist in 1 by Theorem 1.9 and thus n>00 uf lim inf E,) and j.(lim sup E,,) are defined. Now (1(E,) : n € N) isasequencein [0, 00] > and thus liminf z(E,) = lim inf w(E,) and limsupu(E,) = lim sup 2(E,) exist in n-00 n>ook>n n>0o N00 pon [0, 00]. How arej4(liminf E,) and j:(lim sup E,) related respectively to lim inf 4(E,) 7100 noo 100 and lim sup 4(Z,)? The next theorem addresses this question. ROO Theorem 1.28. Let 2 be a measure on a o-algebra A of subsets of a set X. (a) For an arbitrary sequence (E, :n € N) in Q, we have () u(liminf £,) <lim inf CE). n00 n>00 (b) If there exists A € SA with u(A) < 00 such that E, C Aforn €N, then 2) j(lim sup E,,) > lim sup 2(E,). n->co n+>00 (c) both lim E, and lim p(E,) exist, then ROO n->00 @) w( lim Eq) < tim (En). noo n—>00 @ if jim En exist and if there exists A € A with w(A) < 00 such that E, C Aforn €N, then im w(En) exists and 4 Hf lim En) = Jim w(En).
- 46. §1 Measure on a o-algebra of Sets 17 Proof. 1. Recall that lim inf E, = new Mon Be = tim, Chen Ex by the fact that (Neen Ex : 2 € N) is an increasing sequence in %. Then by (a) of Theorem 1.26, we have y(liminf Zn) = lim (Myon Ee) = liminf (Myon Ze) since the limit of a sequence, if it exists, is equal to the limit inferior of the sequence. Since hen Ex C En, we have 14 (Msn Ex) < (En) for n € N by the monotonicity of 2. This then implies lim:inf # (Men Ex) < lim inf u(E,). Continuing the chain of equalities above with this inequality, we have (1). 2. Assume that there exists A € {2 with 4(A) < oo such that E, C A forn € N. Now limsup En = nen Uson Ex = lim Ubon Ex by the fact that (Ubon Ex 2 €N) isa n—00 decreasing sequence in A. Since E, C A for all m € N, we have Ubon E, Cc A for all n & N. Thus we havejz(limsup Ep) = w( lim Uj, Ex) = lim 2 (Upon Ex) by 0) n—>00 no9 RE n>00 I of Theorem 1.26. Now lim (Uren 2x) = lim sup #(Usen Ex) since the limit of a se- quence, if it exists, is equal to the limit superior of the sequence. Then by 5, Zz D En, we have 1 (Ups, Ex) = w(En). Thus lim sup (Upon Ex) = lim sup u(E,). Continuing the chain of equalities above with this jncquality, we have (2). 3. If im, E,, and jim, BA(E,) exist, then im, E,, = liminf E,, and im, BE) = lim nf s.(i,) 80 that (1) reduces to (3) ue “4. If ima, E,, exists, then im sup En = = im, Ey, = lim inf E,. If there exists A € 2 with uA) < oo such that E, C "A for ne N.then by (2) and (1) we have 6) limsup 2(En) = #(lim sup E,} = u( lim Ep) now Aco noo = 2(liminf E,,) < liminf 2(E,). n>00 n>00 Since liminf 4(Z,) < lim sup (£,) the inequalities (5) imply noo n>00 (6) liminf (En) = w{ lim En) = lim sup HE). Thus lim ,(£,) exists and then by (6) we have ( lim E,) = lim (Ey). This proves (4) nora n->00 ni>00 [VII] Measurable Space and Measure Space Definition 1.29. Let & be a o-algebra of subsets of a set X. The pair (X, QW) is called a measurable space. A subset E of X is said to be &-measurable if E € A. Definition 1.30. (a) if 4 is a measure on a o-algebra ‘XA of subsets of a set X, we call the triple (X, A, w) a measure space. (b) A measure pp on a o-algebra A of subsets of a set X is called a finite measure if
- 47. 18 CHAPTER 1 Measure Spaces #(X) < 00. In this case, (X, &, 2) is called a finite measure space. (c) A measure 2 on a o-algebra & of subsets of a set X is called ao -finite measure if there exists a sequence (E, : n € N) in & such that Jen En = X and w(En) < 00 for every n EN. In this case (X, A, ys) is called a o-finite measure space. (d) A set D € & in an arbitrary measure space (X, A, i) is called a o-finite set if there exists a sequence (D, : n € N) in & such that Unen Dy, = D and p(D,) < 00 for every neN. Lemma 1.31. (a) Let (X, 2, 4) be a measure space. If D € Nis a o-finite set, then there exists an increasing sequence (F, :n € N) in Ql such that lim | F, = Dand w(F,) < oofor 1 everyn & N and there exists a disjoint sequence (G, : n € N) in'A such that |J,en Gn = D and 4(Gn) < 00 for everyn EN. (b) If (X, &, 2) is a o-finite measure space then every D € Wis aa-finite set. Proof. 1. Let (X, &, jz) be a measure space. Suppose D € 2 is a o-finite set. Then there exists a sequence (D, : n € N) in & such that |), 2 Dy = D and u(D,) < 00 for every n EN. Foreachn EN, let F, = Ufet D,. Then (F, : n € N) is an increasing sequence in 2 such that jim Fr = Unen Fn = Unen Dn = D and u(F,) = u(Uf-1 De) < Yhe1 H(Dx) < 00 for every n EN. Let G, = Fy and Gy = Fy Ufc& forn > 2. Then (Gq : n € N) is adisjoint sequence in 2 such that L,en Ga = Unser, = D as in the Proof of Lemma 1.21, (G1) = w(Fi) < 00 and 4(Gn) = (Fa Uli Fe) S a(Fa) < 00 forn > 2. This proves (a). 2. Let (X, &, 2) be a o-finite measure space. Then there exists a sequence (EZ, : n € N) in 2 such that Len En = X and u(E,) < 00 for every n € N. Let D € 2. For each n EN, let Dp = DN Ep. Then (D, : n € N) is a sequence in 2 such that L),.4 Dn = D and 4(D,) < 4(E,) < 00 for every n € N. Thus D is a o-finite set. This proves (b). Definition 1.32. Given a measure 4 on a o-algebra A of subsets of a set X. A subset E of X is called a null set with respect to the measure yz if E € Wand u(E) = 0. In this case we say also that E is a nuil set in the measure space (X, A, uz). (Note that 9 is a null set in amy measure space but a null set in a measure space need not be 9.) Observation 1.33. A countable union of null sets in a measure space is a null set of the measure space. Proof. Let (E, : n € N) be a sequence of null sets in a measure space (X, &, 4). Let E = UnenEn- Since & is closed under countable unions, we have E € Q&. By the countable subadditivity of 2 on &, we have u(E) < Yen #(En) = 0. Thus w(E) = 0. This shows that E is a null set in (X, 2, 2). © Definition 1.34. Given a measure yt on a o-algebra & of subsets of a set X. We say that the o-algebra X is complete with respect to the measure js if an arbitrary subset Eo of a null set
- 48. §1 Measure on a o-algebra of Sets 19 E with respect to us is a member of A (and consequently has (Eo) = 0 by the monotonicity of 4). When & is complete with respect to 1, we say that (X, A, 4) is a complete measure space, Example. Let X = {a,b,c}. Then & = {9, {a}, {b, c}, X} is a o-algebra of subsets of X. If we define a set function 4 on X by setting (0) = 0, w({a}) = 1, u({b,c}) = 0, and 4.(X) = 1, then yw is a measure on 2. The set {b, c} is a null set in the measure space (X, A, 2), but its subset {5} is not a member of 2. Therefore (X, A, 2) is not a complete measure space. Definition 1.35. (a) Given a measurable space (X, UM). An A-measurable set E is called an atom of the measurable space if @ and E are the only A-measurable subsets of E. (b) Given a measure space (X, A, 4). An A-measurable set E is called an atom of the measure space if it satisfies the following conditions : 1° 2(E) > 0, 2 Eg C £, Eg € R= p(Eo) =0 or u(Eq) = (EZ). Observe that if Z is an atom of (X, 20) and w(£) > 0, then E is an atom of (X, A, yz). Example. In a measurable space (X, &) where X = {a, b, c} and A = {G, {a}, {b, c}, X}, if we define a set function on 2 by setting (8) = 0, w{{a}) = 1, u({b,c}) = 2, and HX) = 3, then yz is a measure on &. The set {b, c} is an atom of the measure space (X, 2, w). [VI] Measurable Mapping Let f be a mapping of a subset D of a set X into a set Y. We write D(f) and (f) for the domain of definition and the range of f respectively. Thus D(f=DcX, MP) ={y oY: y = f(&) forsomex e D(P)} CY. For the image of D(f) by f we have f(D(f)) = N(f). For an arbitrary subset E of Y we define the preimage of E under the mapping f by f(E) = [x eX: f(x) € E} = {x DY): f@) € E}. Note that F is an arbitrary subset of Y and need not be a subset of S8(f). Indeed E may be disjoint from 9%(f), in which case f—1(E) = 9. In general we have f(f—1(E)) c E. For an arbitrary collection € of subsets of Y, we let f-1(€) := {f-1(Z) : E € €}. Observation 1.36. Given sets X and Y. Let f be a mapping with D(f) C X and
- 49. 20 CHAPTER 1 Measure Spaces SRCf) CY. Let E and Ey be arbitrary subsets of Y. Then (a) £1) =D(), Q) SIE) =f O =f OMIT =DBIN II ®. @Q) f-(E) = (f-"(®)’ _ provided that D(f) = ¥, 4 £7" (Use Bu) = Unea f-" (Ea), 6) I (Muea Ea) = Maea f'(Ea). Proposition 1.37. Given sets X and Y. Let f beamapping withD(f) C X and Rf) c Y. If 33 is ao-algebra of subsets of Y then f—'(98) is a o-algebra of subsets of the set D(f). In particular, if D(f) = X then f—) (9B) is a o-algebra of subsets of the set X. Proof. Let 93 be a o-algebra of subsets of the set Y. To show that f—'(93) is a c-algebra of subsets of the set D(/) we verify: 1° D(f) € f-'B). 2 Ae f-1(B) > D(f)A€ f7'(3). 3° (An: n © N) C f71(93) > Upen An © £713). This is done below. 1. By (1) of Observation 1.36, we have D(f) = f7'(Y) © f-1(9B) since Y € B. 2. Let A € f—1(93). Then A = f—'(B) for some B € 93. Then by (2) of Observation 1.36 we have D(f) A = D(f) f1(B) = f-1(B*). Since B is aa c-algebra, B ¢ B implies B° € %. Then f~!(B°) € f—1(83). This shows that D(f) A € f—1(98). 3. Let (A, :n € N) bea sequence in f—!(93). Then A, = f—!(B,) for some B, € 3 for each n € N. Then by (4) of Observation 1.36, we have Un =U 6) = 7(U Be) € £1), neN neN nen since U,<n Bn € B. Definition 1.38. Given two measurable spaces (X, &) and (Y, %3). Let f be amapping with D(f) c X and K(f) C Y. We say that f is a A/%3-measurable mapping if f—'(B) ¢ A for every B € &, that is, f—1(98) c A. According to Proposition 1.37 for an arbitrary mapping f of D(f) Cc X into Y, f71(8) is a o-algebra of subsets of the set O(f). 2/%-measurability of the mapping f requires that the o-algebra f—1(95) of subsets of D(f) be a subcollection of the o-algebra A of subsets of X. Note alsothat the 21/93-measurability of f implies that D(f) = f—1(Y) « A since Y € %. Therefore, to construct a 21/93-measurable mapping f on a subset D of X we must assume from the outset that D € 2. Observation 1.39. Given two measurable spaces (X, 2) and (Y, 23). Let f be a 2/B- measurable mapping.
- 50. §1 Measure on a o-algebra of Sets 21 (a) If A, is a o-algebra of subsets of X such that AW, > A, then f is 2, /%-measurable. (b) If Bp is a o-algebra of subsets of Y such that 3 C B, then f is M/2o-measurable. Proof. (a) follows from f~1(93) C 2M C My and (b) from f—!(Bo) c f-1(9B) CA. Composition of two measurable mappings is a measurable mapping provided that the two measurable mappings form a chain. To be precise, we have the following: Theorem 1.40. (Chain Rule for Measurable Mappings) Given measurable spaces (X, 2), (¥Y, B), and (Z, €). Let f be a mapping with D(f) Cc X, Rf) C Y, g be a mapping with D(g) Cc Y, (gz) C Z such that R(f) C D(g) so that the composite mapping go f is defined with D(g o f) C X and Rigo f) C Z. If f is A/B-measurable and g is 383 /€-measurable, then g o f is A/€-measurable, Proof. By the 21/%3-measurability of f, we have f—1(93) Cc &, and by the B/¢- measurability of g, we have g1(€) c %. Thus (go f)1(€) = f-(g(©)) c f-(B) CA. a The 2/%3-measurability condition can be reduced when % is the a-algebra generated by acollection € of subsets of Y. Thus we have the following: Theorem 1.41. Given two measurable spaces (X, 2) and (Y¥, 3B), where B = o(€) and € is an arbitrary collection of subsets of Y. Let f be a mapping with D(f) « A and R(f) CY. Then f is a A/B-measurable mapping of D(f) into Yif and only if fI@) cm Proof. If f is a 1/%3-measurable mapping of D(f) into Y, then f—!(93) C A so that f71@) CM. Conversely if f-"(E) Cc A, then o(f—(E)) Cc o(A) = A. Now by Theorem 1.14, o(f—1(€)) = f—(o(€)) = f-1(93). Thus f—1(98) C Wand f isa Mt /%8-measurable mapping of D(f). a Proposition 1.42. Given two measurable spaces (X, A) and (Y, By), where Y is a topo- logical space and ‘By is the Borel c-algebra of subsets of Y. Let f be a mapping with D(f) € A and R(f) CY. Let Oy and €y be respectively the collection of all open sets and the collection of all closed sets in Y. (a) f is a A/By-measurable mapping of Df) into Yif and only if f-(Oy) C A. (b) f is a A/%By-measurable mapping of D(f) into ¥if and only if f—'(€y) C A. Proof. Since By = o(Oy) = o(€y), the Proposition is a particular case of Theorem 141. © Theorem 1.43. Given two measurable spaces (X, Bx) and (Y, By) where X and Y are topological spaces and 3B and By are the Borel o-algebras of subsets of X and Y respectively. If f is a continuous mapping defined on a set D € Sx, then f isa Sx /By-measurable mapping of D into Y.
- 51. 22 CHAPTER 1 Measure Spaces Proof. Let V be an open set in Y. The continuity of f on D implies that f—1(V) = UND where U is an open set in X so that f—!(V) € %3y. Since this holds for every open set V in Y, f is a 3x /My-measurable mapping of D into Y by (a) of Proposition 1.42. A particular case of Theorem 1.43 is when we have a real-valued continuous function f defined on a set D € By where By is the Borel o-algebra of subsets of a topological space X. In this case we have (Y, By) = (R, BR). By Theorem 1.43, f is a Bx/Br- measurable mapping of D into R. [IX] Induction of Measure by Measurable Mapping Let ys be a measure on a o-algebra 2 of subsets of a set X. We show next that a measurable mapping of the measurable space (X, 2) into another measurable space (Y, 33) induces a measure on the o-algebra 93. The induced measure on %B is called the image measure induced by the measurable mapping. Theorem 1.44. (Image Measure) Given two measurable spaces (X, A) and (Y, B). Let f be aSA/B-measurable mapping of X into Y. Let be a measure on XU. The set function defined by vy = 0 f—| on B, that is, v(B) = w (f-1(B)) for B € B, is a measure on B. Proof. Since f is a 21/93-measurable mapping of X into Y, we have f—1(B) € Ml forevery B € ® and then v(B) = » (f—1(B)) € [0, oo]. Also vx) = u(f 1@) = w@ =0. Let (B, : n € N) be a disjoint sequence in %3. Then (f—!(B,) : » € N) is a disjoint sequence in & and f—! (U,c~ Bn) = Unew f1(Bn) € &. Thus we have the equality »(Upen Ba) = #(F7"(Unen Bn) = Dnen #(F'(Bn)) = Caen ¥(Bn)- This shows that v is countably additive on 1. Therefore v is a measure on 3. Problems Prob. 1.1. Given two sequences of subsets (E,, : n € N) and (F, : n € N) of aset X. (a) Show that (1) _sdiminf E,, U liminf F, C liminf(E, U F,) C liminf E, Ulim sup Fy noo AS>O n>00 noo n-P0O c lim sup(E£, U F,) C lim sup E,, U lim sup F,. noo n>00 noo (b) Show that (2) liminf E, M liminf F, C liminf(E, N F,) C liminf E, M lim sup F, 100 m0 100 100 no C lim sup(E, N F,) C lim sup E, NM lim sup F,. nooo n>00 noo
- 52. §1 Measure on a o-algebra of Sets 23 (c) Show thatif lim £, and lim F, exist, then lim (En u Fr) and lim (En n Fy) exist n00 n00 n00 n00 and moreover Q) lim (EZ, U F,) = lim E,U lim Fy, n-00 norco" n-+00 (4) lim (E, Fn) = lim E,A lim Fy. n—->0O noo n—->00 Prob. 1.2. (a) Let (A, : n € N) be a sequence of subsets ofa set X. Let (B, :n € N) bea sequence obtained by dropping finitely many entries in the sequence (A, : n € N). Show that lim inf By = = lim inf An and fim sup Ba = im m sup A,. Show that im, Bp, exists if and onlyif jim, Ag exists andwhen ‘they exist they a ae equal. (b) Let (An:TR € N) and (B, : n € N) be two sequences of subsets of a set X such that A, = 8B, for all but finitely many n € N. Show that fim inf By, = lim inf Ap and lim sup B, = iimn sup Ay. Show that jim, B,, exists if and only’ if“iim A, exists and when a>00 they exist they are re equal. Prob. 1.3. Let (Z,, : n € N) bea disjoint sequence of subsets of a set X. Show that im, En exists and lim £, = 9. n>00 Prob. 1.4. Leta € R and let (x, : n € N) be a sequence of points in R, all distinct from a, such that jim, X, = a. Show that im a {xn} exists and jima {xn} = 9 and thus jim {xn} # {a}. Prob. 1.5. ForE c Randt € R, let us write E+¢ = {x +f €R:x € E} and call it the translate of E by t. Let (f,: n € N) be a strictly decreasing sequence in R such that jim, t, = Oand let E,= E +t, forn € N. Let us investigate the existence of im, En. (a) Let E = (— 00, 0). Show that in, E, = (—00, 0]. (b) Let E = {a} where a € R. Show that Jim, E, =. (c) Let E = [a, b] where a, b € Randa < b. “Show that fim, E, = (a, 6]. (d) Let E = (a, b) where a, b € Randa < b. Show that jim, En = (a, b]. (e) Let E = Q, the set of all rational numbers. Assume ‘that (i,: n € N) satisfies the additional condition that £, is a rational number for all but finitely many n € N. Show that Jim, E,= E. Ol Let E = Qas in (d) but assume that (1,: n € N) satisfies the additional condition that t, is a rational number for infinitely many n € N and 1, is an irrational number for infinitely many n € N. Show that Jima, E,, does not exist. Prob. 1.6. The characteristic function 14 of a subset A of a set X is a function on X defined by 1 forx € A, 1a@) = { 0 forx € A‘. Let (A, : n € N) be a sequence of subsets of X and A be a subset of X.
- 53. 24 CHAPTER 1 Measure Spaces (a) Show that if jim, An = A then lim 14, = 1, on X. 100 {b) Show that if ‘im 14, =1, on X then jim, Ayn = A. Prob. 1.7. Let 2 be a o-algebra of subsets of a set X and let Y be an arbitrary subset of X. Let 3 = {ANY : A € &}. Show that % is a o-algebra of subsets of Y. Prob. 1.8. Let 21 be a collection of subsets of a set X with the following properties: 1°, XeA, 2°. A BEA>DAB=ANB EA, Show that 2 is an algebra of subsets of the set X. Prob. 1.9. Let 2 be an algebra of subsets of a set X. Suppose 2 has the property that for every increasing sequence (A, : n € N) in 2, we have L),-w An € A. Show that & is a o-algebra of subsets of the set X. Prob. 1.10. Let (X, 2l) be a measurable space and let (Z, : n € N) be an increasing sequence in % such that J, oy En = X. (a) Let A, = AN Ep, that is, Uy, = {AN E, : A € A}. Show that , is a oc-algebra of subsets of E, for each n € N. (b) Does Jen Mn = A hold? Prob. 1.11. (a) Show that if (2, : n € N) is an increasing sequence of algebras of subsets of a set X, then _), cv 2, is an algebra of subsets of X. (b) Show that if (1, : n € N) is a decreasing sequence of algebras of subsets of a set X, then (cy Mn is an algebra of subsets of X. Prob. 1.12. Let (X, 21) be a measurable space. Let us call an 2{-measurable subset E of X an atom in the measurable space (X, 2) ifE ~ @ and G and E are the only 2-measurable subsets of E. Show that if £, and E> are two distinct atoms in (X, MA) then they are disjoint. Prob. 1.13. For an arbitrary collection € of subsets of a set X, let a(€) be the algebra generated by &, that is, the smallest algebra of subsets of X containing €, and let o(€) be the o-algebra generated by €. Prove the following statements: @) n(ate) = =a(€), (b) o(o(€))= o(€), () a(€) Cc o(€), (d) if € is a finite collection, then a(€) = o(€), (©) o(a(€)) =o(€). (Hint for (d): Use Prob. 1.18 below.) Prob. 1.14. Let (A,: n € N) be a monotone sequence of c-algebras of subsets of a set X and let A = lim, An. {a) Show that if (ln : n € N) is a decreasing sequence then 2 is a o-algebra. (b) Show that if (2, : n € N) is an increasing sequence then 2 is an algebra but 2 may not be a o-algebra by constructing an example. Prob. 1.15. Let € = {Aj, ..., An} be a disjoint collection of nonempty subsets of a set X
- 54. §1 Measure on a o-algebra of Sets 25 such that |_7_, Ay = X. Let ¥ be the collection of all arbitrary unions of members of €. (a) Show that ¥ = o(€), the smallest o-algebra of subsets of X containing €. (b) Show that the cardinality of o (€) is equal to 2". Prob. 1.16. Let € = {A; : i € N} be a disjoint collection of nonempty subsets of a set X such that |_),<9y Ai = X. Let ¥ be the collection of all arbitrary unions of members of €. (a) Show that ¥ = o(€), the smallest o-algebra of subsets of X containing €. (b) Show that the cardinality of o (€) is equal to 2%, Prob. 1.17. Show that a o-algebra of subsets of a set cannot be a countably infinite collection, that is, it is either a finite or an uncountable collection. Prob. 1.18. Let € = {£1,--- , E,} be a finite collection of distinct, but not necessarily disjoint, subsets of a set X. Let D be the collection of all subsets of X of the type: Ay A EL Ey? n+. Ee, where A; assumes the values {1, 0} and E} = E; and E° = Ef fori = 1,...,n. Let ¥ be the collection of all arbitrary unions of members of D. (a) Show that any two distinct members of D are necessarily disjoint, that is, D is a disjoint collection. (b) Show that the cardinality of D is at most 2”. (c) Show that ¥ = a(€). (d) Show that the cardinality of #(€) has at most 2 (e) Show that o(€) = a(€). Remark. For an arbitrary collection € of subsets of a set X, the smallest o-algebra of subsets of X containing €, o(€), always exists according to Theorem 1.11. Prob. 1.18 presents a method of constructing a(€) for the case that € is a finite collection. Prob. 1.19. Let € be an arbitrary collection of subsets of a set X. Consider a(€), the smallest algebra of subsets of X containing €. Show that for every A € a(€) there exists a finite subcollection €4 of € depending on A such that A € a(€,). Prob. 1.20. Let € be an arbitrary collection of subsets of a set X. Consider o(€), the smallest a-algebra of subsets of X containing €. Show that for every A € o(€) there exists an at most countable subcollection €4 of € depending on A such that A € o(€,). Prob. 1.21. Let yz be a measure on a o-algebra A of subsets of a set X and let Wy be a sub-c-algebra of 2, that is, Xp is a o-algebra of subsets of X and Wo C MA. Show that the restriction of jz to Ao is a measure on Wo. Prob. 1.22. Let (X, A, 2) be a measure space. Show that for any £1, Ez € MU we have the equality: (Ey U Ey) + w(E1 9 £2) = w(E1) + w(E2). Prob. 1.23, Let (X, 2) be a measurable space. Let yzbe a measure on the a-algebra 2& of subsets of X and let a, > 0 for every k € N. Define a set function yz on 2 by setting = Deen Okie. Show that us is a measure on 2. Prob. 1.24, Let X = (0, oc) and let J = {4 : k € N} where , = (K—1,k] fork EN.
- 55. Discovering Diverse Content Through Random Scribd Documents
- 59. The Project Gutenberg eBook of Descriptive Analyses of Piano Works
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Descriptive Analyses of Piano Works Author: Edward Baxter Perry Release date: February 14, 2014 [eBook #44910] Most recently updated: October 24, 2024 Language: English Credits: Produced by Sean (scribe_for_hire@yahoo.com), based on page images generously made available by the Internet Archive (https://archive.org/details/descriptiveanaly00perriala). *** START OF THE PROJECT GUTENBERG EBOOK DESCRIPTIVE ANALYSES OF PIANO WORKS ***
- 61. DESCRIPTIVE ANALYSES OF PIANO WORKS FOR THE USE OF TEACHERS, PLAYERS, AND MUSIC CLUBS BY EDWARD BAXTER PERRY PHILADELPHIA THEODORE PRESSER CO. LONDON, WEEKES & CO. Copyright, 1902, by Theodore Presser International Copyright Printed in the United States of America
- 62. My Keys I. To no crag-crowning castle above the wild main, To no bower of fair lady or villa in Spain; To no deep, hidden vaults where the stored jewels shine, Or the South’s ruddy sunlight is prisoned in wine; To no gardens enchanted where nightingales sing, And the flowers of all climes breathe perpetual spring: To none of all these They give access, my keys, My magical ebon and ivory keys. II. But to temples sublime, where music is prayer, To the bower of a goddess supernally fair; To the crypts where the ages their mysteries keep, Where the sorrows and joys of earth’s greatest ones sleep; Where the wine of emotion a life’s thirst may still, And the jewels of thought gleam to light at my will: To more than all these They give access, my keys, My magical ebon and ivory keys. III.
- 63. To bright dreams of the past in locked cells of the mind, To the tombs of dead joys in their beauty enshrined; To the chambers where love’s recollections are stored, And the fanes where devotion’s best homage is poured; To the cloudland of hope, where the dull mist of tears As the rainbow of promise illumined appears; To all these, when I please, They give access, my keys, My magical ebon and ivory keys. Only an Interpreter The world will still go on the very same When the last feeble echo of my name Has died from out men’s listless hearts and ears These many years. Its tides will roll, its suns will rise and set, When mine, through twilight portals of regret, Has passed to quench its pallid, parting light In rayless night, While o’er my place oblivion’s tide will sweep To whelm my deeds in silence dark and deep, The triumphs and the failures, ill and good, Beneath its flood.
- 64. Then other, abler men will serve the Art I strove to serve with singleness of heart; Will wear her thorned laurels on the brow, As I do now. I shall not care to ask whose fame is first, Or feel the fever of that burning thirst To win her warmest smile, nor count the cost Whate’er be lost. As I have striven, they will strive to rise To hopeless heights, where that elusive prize, The unattainable ideal, gleams Through waking dreams. But I shall sleep, a sleep secure, profound, Beyond the reach of blame, or plaudits’ sound; And who stands high, who low, I shall not know: ’Tis better so. For what the gain of all my toilsome years, Of all my ceaseless struggles, secret tears? My best, more brief than frailest summer flower, Dies with the hour. My most enduring triumphs swifter pass Than fairy frost-wreaths from the window glass: The master but of moments may not claim A deathless name. Mine but the task to lift, a little space, The mystic veil from beauty’s radiant face That other men may joy thereon to see, Forgetting me.
- 65. Not mine the genius to create the forms Which stand serenely strong, thro’ suns and storms, While passing ages praise that power sublime Defying time. Mine but the transient service of a day, Scant praise, too ready blame, and meager pay: No matter, though with hunger at the heart I did my part. I dare not call my labor all in vain, If I but voice anew one lofty strain: The faithful echo of a noble thought With good is fraught. For some it cheers upon life’s weary road, And some hearts lightens of their bitter load, Which might have missed the message in the din Of strife and sin. My lavished life-blood warmed and woke again The still, pale children of another’s brain, Brimmed full the forms which else were cold, Tho’ fair of mold. And thro’ their lips my spirit spoke to men Of higher hopes, of courage under pain, Of worthy aspirations, fearless flight To reach the light. Then, soul of mine, content thee with thy fate, Though noble niche of fame and guerdon great Be not for thee: thy modest task was sweet At beauty’s feet.
- 66. The Artist passes like a swift-blown breeze, Or vapors floating up from summer seas; But Art endures as long as life and love: For her I strove. Contents PAGE Introduction, 11 Esthetic versus Structural Analysis, 15 Sources of Information Concerning Musical Compositions, 23 Traditional Beethoven Playing, 32 Beethoven: The Moonlight Sonata, Op. 27, No. 2, 45 Beethoven: Sonata Pathétique, Op. 13, 50 Beethoven: Sonata in A Flat Major, Op. 26, 55 Beethoven: Sonata in D Minor, Op. 31, No. 2, 61 Beethoven: Sonata in C Major, Op. 53, 64 Beethoven: Sonata in E Minor, Op. 90, 68 Beethoven: Music to “The Ruins of Athens,” 72 Weber: Invitation to the Dance, Op. 65, 81 Weber: Rondo in E Flat, Op. 62, 86 Weber: Concertstück, in F Minor, Op. 79, 90 Weber-Kullak: Lützow’s Wilde Jagd, Op. 111, No. 4, 93 Schubert: (Impromptu in B Flat) Theme and Variations, Op. 142, No. 3, 99 Emotion in Music, 105 Chopin: Sonata, B Flat, Op. 35, 113 The Chopin Ballades, 118
- 67. Chopin: Ballade in G Minor, Op. 23, 123 Chopin: Ballade in F Major, Op. 38, 130 Chopin: Ballade in A Flat, Op. 47, 137 Chopin: Polonaise, A Flat Major, Op. 53, 142 Chopin: Impromptu in A Flat, Op. 29, 147 Chopin: Fantasie Impromptu, Op. 66, 149 Chopin: Tarantelle, A Flat, Op. 43, 152 Chopin: Berceuse, Op. 57, 156 Chopin: Scherzo in B Flat Minor, Op. 31, 158 Chopin: Prelude, Op. 28, No. 15, 161 Chopin: Waltz, A Flat, Op. 42, 168 Chopin’s Nocturnes, 172 Chopin: Nocturne in E Flat, Op. 9, No. 2, 174 Chopin: Nocturne, Op. 27, No. 2, 176 Chopin: Nocturne, Op. 32, No. 1, 179 Chopin: Nocturne, Op. 37, No. 1, 183 Chopin: Nocturne, Op. 37, No. 2, 186 Chopin: Polish Songs, Transcribed for Piano by Franz Liszt, 191 Liszt: Poetic and Religious Harmonies, No. 3, Book 2, 194 Liszt: First Ballade, 199 Liszt: Second Ballade, 201 Transcriptions for the Piano by Liszt, 203 Wagner-Liszt: Spinning Song from “The Flying Dutchman,” 205 Wagner-Liszt: Tannhäuser March, 208 Wagner-Liszt: Abendstern, 209 Wagner-Liszt: Isolde’s Love Death, 210 Schubert-Liszt: Der Erlkönig, 213 Schubert-Liszt: Hark! Hark! the Lark, 216 Schubert-Liszt: Gretchen am Spinnrad, 217 Liszt: La Gondoliera, 219
- 68. The Music of the Gipsies and Liszt’s Hungarian Rhapsodies, 222 Rubinstein: Barcarolle, G Major, 237 Rubinstein: Kamennoi-Ostrow, No. 22, 241 Grieg: Peer Gynt Suite, Op. 46, 247 Grieg: An den Frühling, Op. 43, No. 6, 257 Grieg: Vöglein, Op. 43, No. 4, 260 Grieg: Berceuse, Op. 38, No. 1, 261 Grieg: The Bridal Procession, from “Aus dem Volksleben,” Op. 19, No. 2, 264 Saint-Saëns: Le Rouet d’Omphale, 271 Saint-Saëns: Danse Macabre, 276 Counterparts among Poets and Musicians, 281 DESCRIPTIVE ANALYSES OF PIANO WORKS Introduction The material comprised in the following pages has been collected for use in book form by the advice and at the earnest request of the publisher, as well as of many musical friends, who express the belief that it is of sufficient value and interest to merit a certain degree of permanency, and will prove of practical aid to teachers and students of music. A portion of it has already appeared in print in the program books of the Derthick Musical Literary Society and in different musical journals; and nearly all of it has been used at various times in my own Lecture Recitals.
- 69. The book is merely a compilation of what have seemed the most interesting and valuable results of my thought, reading, and research in connection with my Lecture Recital work during the past twenty years. In the intensely busy life of a concert pianist a systematic and exhaustive study of the whole broad field of piano literature has been utterly impossible. That would require the exclusive devotion of a lifetime at least. My efforts have been necessarily confined strictly to such compositions as came under my immediate attention in connection with my own work as player. The effect is a seemingly desultory and haphazard method in the study, and an inadequacy and incoherency in the collective result, which no one can possibly realize or deplore so fully as myself. Still the work is a beginning, a first pioneer venture into a realm which I believe to be not only new, but rich and important. I can only hope that the example may prompt others, with more leisure and ability, to follow in the path I have blazed, to more extensive explorations and more complete results. Well-read musicians will find in these pages much that they have learned before from various scattered sources. Naturally so. I have not originated my facts or invented my legends. They are common property for all who will but seek. I have merely collected, arranged, and, in many instances, translated them into English. I claim no monopoly. On the other hand, they may find some things they have not previously known. In such cases I venture to suggest to the critically and incredulously inclined, that this does not prove their inaccuracy, though some have seemed to fancy that it did. Not to know a thing does not always conclusively demonstrate that it is not so. To the general reader let me say that this book represents the best thought and effort of my professionally unoccupied hours during the past twenty years. It comes to you with my heart in it, bringing the wish that the material here collected may be to you as interesting and helpful as it has been to me in the gathering. The actual writing
- 70. has mainly been done on trains, or in lonely hotel rooms far from books of reference, or aids of any kind; so occasional inexactitudes of data or detail are by no means improbable, when my only resource was the memory of something read, or of personal conversation often years before. With the limited time at my disposal, a detailed revision is not practicable, and I therefore present the articles as originally written. Take and use what seems of value, and the rest pass by. The plan and purpose of the book rest simply upon the theory that the true interpretation of music depends not only on the player’s possession of a correct insight into the form and harmonic structure of a given composition, but also on the fullest obtainable knowledge concerning the circumstances and environment of its origin, and the conditions governing the composer’s life at the time, as well as any historical or legendary matter which may have served him as inspiration or suggestion. My reason for now presenting it to the public is the same as that which has caused me to devote my professional life exclusively to the Lecture Recital—namely, because experience has proved to me that a knowledge of the poetic and dramatic content of a musical work is of immense value to the player in interpretation, and to the listener in comprehension and enjoyment of any composition, and because, except in scattered fragments, no information of just this character exists elsewhere in print. It being, as explained, impossible to make this collection of analyses complete, or even approximately so, it has seemed wise to limit the number here included to just fifty, so as to keep the book to a convenient size. I have endeavored to select those covering as large a range and variety as possible, with the view of making them as broadly helpful and suggestive as may be. It is my intention to continue my labors along this line so far as strength and opportunity permit, in the faith that I can devote my efforts to no more useful end.
- 71. Edward Baxter Perry. Esthetic versus Structural Analysis It has been, and still is, the general custom among most musicians, when called upon to analyze a composition for the enlightenment of students or the public, or in the effort to broaden the interest in their art, to think and speak solely of the form, the structure of the work, to treat it scientifically, anatomically—to dwell with sonorous unction upon the technical names for its various divisions, to lay bare and delightedly call attention to its neatly fashioned joints, to dilate upon the beauty of its symmetrical proportions, and show how one part fits into or is developed out of another—in brief, to explain more or less intelligently the details of its mechanical construction, without a hint or a thought as to why it was made at all, or why it should be allowed to exist. With the specialist’s engrossing absorption in the technicalities of his vocation, they expect others to share their interest, and are surprised and indignant to find that they do not. They forget that to the average hearer this learned dissertation upon primary and secondary subjects, episodical passages, modulation to related and unrelated keys, cadences, return of the first theme, etc., has about as much meaning and importance as so much Sanskrit. It is well enough, so far as it goes, in the classroom, where students are being trained for specialists, and need that kind of information; but it is only one side,—the mechanical side,—and the general public needs something else; and even the student, however gifted, if he is to become more than a mere technician, must have something else; for composition and interpretation both have their mere technic, as much as keyboard manipulation, which is, however, only the means, not the end. Knowledge of and insight into musical form are necessary to the player, but not to the listener, even for the highest artistic
- 72. appreciation and enjoyment, just as the knowledge of colors and their combination is essential to the painter, but not to the beholder. The poet must understand syntax and prosody, the technic of rhyme-making and verse-formation; but how many of his readers could analyze correctly from that standpoint the poem they so much enjoy, or give the scientific names for the literary devices employed? Or how many of them would care to hear it done, or be the better for it if they did? The public expects results, not rules or formulas; effects, not explanations of stage machinery; food and stimulus for the intellect, the emotions, the imagination, not recipes of how they are prepared. The value of esthetic analysis is undeniably great in rendering this food and stimulus, contained in every good composition, more easily accessible and more readily assimilated, by a judicious selection and partial predigestion, so to speak, of the different artistic elements in a given work, and a certain preparation of the listener to receive them. This is, of course, especially true in the case of the young, and those of more advanced years, to whom, owing to lack of training and opportunity, musical forms of expression are somewhat unfamiliar; or, in other words, those to whom the musical idiom is still more or less strange. But there are also very many musicians of established position who are sorely in need of something of the kind to awaken them to a perception of other factors in musical art besides sensuous beauty and the display of skill; to develop their imaginative and poetic faculties, in which both their playing and theories prove them to be deficient; and the more loudly they cry against it as useless and illegitimate, the more palpably self-evident becomes their own crying need of it. Esthetic analysis consists in grasping clearly the essential artistic significance of a composition, its emotional or descriptive content, either with or without the aid of definite knowledge concerning the circumstances of its origin, and expressing it plainly in a few simple, well-chosen words, comprehensible by the veriest child in music, whether young or old in years, conveying in a direct, unmistakable, and concrete form the same general impressions which the
- 73. composition, through all its elaborations and embellishments, all its manifold collateral suggestions, is intended to convey, giving a skeleton, not of its form, but of its subject-matter, so distinctly articulated that the most untrained perceptions shall be able to recognize to what genus it belongs. Of course, when it is possible, as it is in many cases, to obtain and give reliable data concerning the conception and birth of a musical work, the actual historical or traditional material, or the personal experience, which furnished its inspiration, the impulse which led to its creation, it is of great assistance and value; and this is especially so when the work is distinctly descriptive of external scenes or human actions. For example, take the Schubert-Liszt “Erlkönig.” Here the elements embodied are those of tempest and gloom, of shuddering terror, of eager pursuit and panic-stricken flight, ending in sudden, surprised despair. These may be vaguely felt by the listener when the piece is played, with varying intensity according to his musical susceptibility; but if the legend of the “Erlkönig,” or “Elf- king,” is narrated and attention directly called to the various descriptive features of the work,—the gallop of the horse, the rush and roar of the tempest through the depths of the Black Forest, the seductive insistence and relentless pursuit of the elf-king, the father’s mad flight, the shriek of the child, and the final tragic ending, all so distinctly suggested in the music,—the impression is intensified tenfold, rendered more precise and definite; and the undefined sensations produced by the music are focused at once into a positive, complete, artistic effect. Who can doubt that this is an infinite gain to the listener and to art? Again, take an instance selected from a large number of compositions which are purely emotional, with no kind of realistic reference to nature or action, the Revolutionary Etude, by Chopin, Opus 10, No. 12. The emotional elements here expressed are fierce indignation, vain but desperate struggle, wrathful despair. These are easily recognized by the trained esthetic sense. Indeed, the work cannot be properly rendered by one who does not feel them in playing it; and they can be eloquently described in a general way by
- 74. one possessing a little gift of language and some imagination; but many persons find it hard to grasp abstract emotions without a definite assignable cause for them, and are incalculably aided if told that the study was written as the expression of Chopin’s feelings, and those of every Polish patriot, on receipt of the news that Warsaw had been taken and sacked by the Russians. Where such data cannot be found concerning a composition, one can make the content of a work fairly clear by means of description, of analogy and comparison, by the use of poetic metaphor and simile, by little imaginative word-pictures, embodying the same general impression; by any means, in short,—any and all are legitimate,—which will produce the desired result, namely: to concentrate the attention of the student or the listener on the most important elements in a composition, to show him what to listen for and what to expect; to prepare him fully to receive and respond to the proper impression, to tune up his esthetic nature to the required key, so it may re-echo the harmonious soul-utterances of the Master, as the horn-player breathes through his instrument before using it, to warm it, to bring it up to pitch, to put it in the right vibratory condition. The plan of esthetic analysis, in more or less complete form, was used by nearly all of the great teachers, such as Liszt, Kullak, Frau Schumann, and others, and was a very important factor in their instruction. It was used by all the great writers on music who were at the same time eminent musicians, like Liszt, Schumann, Mendelssohn, Mozart, Wagner, Berlioz, Ehrlich, and many more. Surely, with such examples as precedents, not to mention other good and sufficient grounds, we may feel safe in pursuing it to the best of our ability, in print, in the teaching-room, in the concert-hall, whenever and wherever it will contribute to the increase of general musical interest and intelligence, in spite of the outcries of the so- called “purists,” who see and would have us see in musical art only sensuous beauty and the perfection of form, with possibly the addition of, as they might put it, a certain ethereal, spiritual, indefinable something, too sacred to be talked about, too
- 75. transcendental to be expressed in language, too lofty and pure to be degraded to the level of human speech. Who, I ask, are the sentimentalists—they, or we who believe that music, like every other art, is expression, the embodying of human experiences, than which there is no grander or loftier theme on this earth? Trust me, it is not music nor its subject-matter that is nebulous, indistinct, hazy; but the mental conceptions of too many who deal with it. If art is expression, as estheticians agree, and music is an art, as we claim, then it must express something; and, given sufficient intelligence, training, and insight, that something—the vital essence of every good composition—can be stated in words. Not always adequately, I grant, but at least intelligibly, as a key to the fuller, more complex expression of the music; serving precisely like the synopsis to an opera, or the descriptive catalogue in a picture gallery. This is the aim and substance of esthetic analysis. Musicians are many who see in their mistress But physical beauty of “color” and “form,” Who hear in her voice but a sensuous sweetness, No thrill of the heart that is living and warm. They judge of her worth by “perfection of outline,” “Proportion of parts” as they blend in the whole, “Symmetrical structure,” and “finish of detail”; They see but the body—ignoring the soul. She speaks, but they seem not to master her meaning, They catch but the “rhythmical ring of the phrase.”
- 76. She sings, but they dream not a message is borne on The breath of the sigh, while its “cadence” they praise. Her saddest laments are “melodious minors” To them, and her jests are but “notes marked staccato”; Her tenderest pleadings but “themes well developed,” Her rage—but “a climax of chords animato.” In vain she endeavors to rouse their perceptions By touching their brows with her soul-stirring hand They measure her fingers, their fairness admire, Declare her “divine,” but will not understand. Away with such worthless and sense-prompted service; Forgetting the goddess, to worship the shrine; Forgetting the bride, to admire her costume, Her garments that glitter, and jewels that shine: And give us the artists of true inspiration, Whose insight is clear, and whose brains comprehend, To interpret the silver-tongued message of music That speaks to the heart, like the voice of a friend; That wakens the soul to the joys that are higher And purer than all that the senses can give,
- 77. That teaches the language of lofty endeavor, And hints of a life that ’twere worthy to live! For music is Art, and all Art is expression, The “beauty of form” but embodies the thought, Imprisons one ray of that wisdom supernal Which Genius to sense-blinded mortals has brought. Then give us the artist whose selfless devotion To Art and her service is earnest and true, To read us the mystical meaning of music; Musicians are many, but artists are few. Sources of Information Concerning Musical Compositions During my professional career I have received scores of letters from musical persons all over the country, asking for the name of the book or books from which I derive the information, anecdote, and poetic suggestion, concerning the compositions used in my Lecture Recitals, particularly the points bearing upon the descriptive and emotional significance of such compositions. All realize the importance and value of this phase of interpretative work, and many are anxious to introduce it in their teaching or public performances; but all alike, myself not excepted, find the sources of such information scanty and difficult of access. First, let me say frankly that there is no such book, or collection of books. My own meager stock of available material in this line has been laboriously collected, without definite method, and at first without distinct purpose, during many years of extensive
- 78. miscellaneous reading in English, French, and German; supplemented by a rather wide acquaintance among musicians and composers, and the life-long habit of seizing and magnifying the poetic or dramatic bearing and import of every scene, situation, and anecdote. If asked to enumerate the sources from which points of value concerning musical works can be derived, I should answer that they are three, not all equally promising, but from each of which I myself have obtained help, and all of which I should try before deserting the field. These are: First, and perhaps the most important, reading. Second, a large acquaintance among musicians, and frequent conversations with them on musical subjects. Third, an intuitive perception, partly inborn and partly acquired, of the analogies between musical ideas, on the one hand, and the experiences of life and the emotions of the human soul, on the other. I will now elaborate each of these a little, to make my meaning more clear. While there is no book in which information concerning the meaning of musical compositions is collected and classified for convenient reference, such information is scattered thinly and unevenly throughout all literatures,—a grain here, a nugget there, like gold through the secret veins of the earth,—and can be had only by much digging and careful sifting. Now and again you come upon a single volume, like a rich though limited pocket of precious ore, and rejoice with exceeding gladness at the discovery of a treasure. But unfortunately, there is usually nothing in the appearance or nature of such a book to indicate to the seeker before perusal that this treasure is within, or to distinguish it from scores of barren volumes. And the very item of which he may be in search is very likely not here to be found; so he must turn again to the quest, which is much like seeking a needle in a hay-mow, or a pearl somewhere at the bottom of the Indian Ocean. Musical histories, biographies, and essays—what is usually termed distinctly musical literature—by no means exhibit the only productive soil, though they are certainly the most fruitful, and should be first
- 79. turned to, because nearest at hand. Poetry, fiction, travels, personal reminiscences, in short every department of literature, from the philosophy of Schopenhauer to the novels of George Sand, must be made to contribute what it can to the stock of general and comprehensive knowledge, which is our ambition. I instance these two authors, because, while neither of them wrote a single work which would be found embraced in a catalogue of musical literature, the metaphysical speculations of Schopenhauer are known to have had great influence upon Wagner’s personality, and through that, of course, upon his music; while in some of the characteristics of George Sand will be found the key to certain of Chopin’s moods, and their musical expression. But even where no such relation between author and composer can be traced, I deem one could rarely read a good literary work, chosen at random, without chancing upon some item of interest or information, which would prove directly or indirectly of value to the professional musician in his life-work. And this is entirely apart from the general broadening, developing, and maturing influence of good reading upon the mind and imagination, which may be added to the more direct benefit sought, forming a background of esthetic suggestion and perception, against which the beauties of tone-pictures stand forth with enhanced power and heightened color. I know of no better plan to suggest to those striving for an intelligent comprehension of the composer’s meaning in his great works than much and careful reading of the best books in all departments, and the more varied and comprehensive their scope the better. In the search for enlightenment concerning any one particular composition, I should advise the student to begin with works, if such exist, from the pen of the composer himself, followed by biographies and all essays, criticisms, and dissertations upon his compositions which are in print. If these fail to give information, he should proceed to read as much as possible regarding the composer’s country and contemporaries, and concerning any and all subjects in which he has become aware, by the study of his life, that the master was interested. The chances are that he will come upon
- 80. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com