**Unformatted text preview: **MEASURE THEORY
Volume 2 D.H.Fremlin By the same author:
Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.
Consequences of Martin’s Axiom, Cambridge University Press, 1982.
Companions to the present volume:
Measure Theory, vol. 1, Torres Fremlin, 2000.
Measure Theory, vol. 3, Torres Fremlin, 2002. First printing May 2001
Second printing April 2003 MEASURE THEORY
Volume 2
Broad Foundations D.H.Fremlin
Reader in Mathematics, University of Essex Dedicated by the Author
to the Publisher This book may be ordered from the publisher at the address below. For price and means of payment see the author’s Web page ,
or enquire from [email protected] First published in 2001
by Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, England
c D.H.Fremlin 2001
∞
The right of D.H.Fremlin to be identified as author of this work has been asserted in accordance with
the Copyright, Designs and Patents Act 1988. This work is issued under the terms of the Design Science
License as published in . For the source files see .
ac.uk/maths/staff/fremlin/mt2.2003/index.htm.
Library of Congress classification QA312.F72
AMS 2000 classification 28A99
ISBN 0-9538129-2-8
Typeset by AMS-TEX Printed in England by Biddles Short Run Books, King’s Lynn 5 Contents
General Introduction 9 Introduction to Volume 2 10 *Chapter 21: Taxonomy of measure spaces
Introduction
211 Definitions 12
12 Complete, totally finite, æ-finite, strictly localizable, semi-finite, localizable, locally determined measure
spaces; atoms; elementary relationships; countable-cocountable measures. 212 Complete spaces 18 Measurable and integrable functions on complete spaces; completion of a measure. 213 Semi-finite, locally determined and localizable spaces 24 Integration on semi-finite spaces; c.l.d. versions; measurable envelopes; characterizing localizability, strict
localizability, æ-finiteness. 214 Subspaces 36 Subspace measures on arbitrary subsets; integration; direct sums of measure spaces. 215 æ-finite spaces and the principle of exhaustion 43 The principle of exhaustion; characterizations of æ-finiteness; the intermediate value theorem for atomless
measures. *216 Examples 47 A complete localizable non-locally-determined space; a complete locally determined non-localizable
space; a complete locally determined localizable space which is not strictly localizable. Chapter 22: The fundamental theorem of calculus
Introduction
221 Vitali’s theorem in R 54
54 Vitali’s theorem for intervals in R. 222 Di↵erentiating an indefinite integral
Monotonic functions are di↵erentiable a.e., and their derivatives are integrable; 223 Lebesgue’s density theorems
f (x) = limh#0
of a function. 1
h R x+h
x f a.e. (x); density points; limh#0 224 Functions of bounded variation 1
2h R x+h
x°h d
dx Rx
a 57
f = f a.e. 63 |f ° f (x)| = 0 a.e. (x); the Lebesgue set Variation of a function; di↵erences of monotonic functions;
sums and products, limits, continuity and
R
di↵erentiability for b.v. functions; an inequality for f £g. 225 Absolutely continuous functions 68
77 Absolute continuity of indefinite integrals; absolutely continuous functions on R; integration by parts;
lower semi-continuous functions; *direct images of negligible sets; the Cantor function. *226 The Lebesgue decomposition of a function of bounded variation 87 Sums over arbitrary index sets; saltus functions; the Lebesgue decomposition. Chapter 23: The Radon-Nikod´
ym theorem
Introduction
231 Countably additive functionals 95
95 Additive and countably additive functionals; Jordan and Hahn decompositions. 232 The Radon-Nikod´
ym theorem 100 Absolutely and truly continuous additive functionals; truly continuous functionals are indefinite integrals;
*the Lebesgue decomposition of a countably additive functional. 233 Conditional expectations 109 æ-subalgebras; conditional expectations of integrable functions; convex functions; Jensen’s inequality. 234 Indefinite-integral measures 117 Measures f µ and their basic properties. 235 MeasurableR transformations
R The formula g(y)∫(dy) = J(x)g(¡(x))µ(dx); detailed conditions of applicability; inverse-measurepreserving functions; the image measure catastrophe; using the Radon-Nikod´
ym theorem. 121 6 Chapter 24: Function spaces
Introduction 133 241 L0 and L0 133 The linear, order and multiplicative structure of L0 ; action of Borel functions; Dedekind completeness
and localizability. 242 L1 141 The normed lattice L1 ; integration as a linear functional; completeness and Dedekind completeness; the
Radon-Nikod´
ym theorem and conditional expectations; convex functions; dense subspaces. 243 L1 152 244 Lp 160 The normed lattice L1 ; completeness; the duality between L1 and L1 ; localizability, Dedekind completeness and the identification L1 ª
= (L1 )§ .
The normed lattices Lp , for 1 < p < 1; H¨
older’s inequality; completeness and Dedekind completeness;
(Lp )§ ª
= Lq ; conditional expectations. 245 Convergence in measure 173 The topology of (local) convergence in measure on L0 ; pointwise convergence; localizability and Dedekind completeness; embedding Lp in L0 ; k k1 -convergence and convergence in measure; æ-finite spaces,
metrizability and sequential convergence. 246 Uniform integrability 184 Uniformly integrable sets in L1 and L1 ; elementary properties; disjoint-sequence characterizations; k k1
and convergence in measure on uniformly integrable sets. 247 Weak compactness in L1 193 A subset of L1 is uniformly integrable i↵ it is relatively weakly compact. Chapter 25: Product measures
Introduction
251 Finite products 199
199 Primitive and c.l.d. products; basic properties; Lebesgue measure on R r+s as a product measure; products of direct sums and subspaces; c.l.d. versions. 252 Fubini’s
theorem
RR
When
Rr . f (x, y)dxdy and 253 Tensor products R 215
f (x, y)d(x, y) are equal; measures of ordinate sets; *the volume of a ball in 233 L1 (µ£∫) as a completion of L1 (µ)≠L1 (∫); bounded bilinear maps; the ordering of L1 (µ£∫); conditional
expectations. 254 Infinite products 244 Products of arbitrary families of probability spaces; basic properties; inverse-measure-preserving functions; usual measure on {0, 1}I ; {0, 1}N isomorphic, as measure space, to [0, 1]; subspaces of full outer
measure; sets determined by coordinates in a subset of the index set; generalized associative law for products of measures; subproducts as image measures; factoring functions through subproducts; conditional
expectations on subalgebras corresponding to subproducts. 255 Convolutions of functions 263 256 Radon measures on R r 275 257 Convolutions of measures 284 R
R
Shifts in R 2 as measure space automorphisms; convolutions of functions on R; h £ (f § g) = h(x +
y)f (x)g(y)d(x, y); f § (g § h) = (f § g) § h; kf § gk1 ∑ kf k1 kgk1 ; the groups R r and ]°º, º].
Definition of Radon measures on R r ; completions of Borel measures; Lusin measurability; image measures; products of two Radon measures; semi-continuous functions.
R
RR
Convolution of totally finite Radon measures on R r ; h d(∫1 §∫2 ) = h(x+y)∫1 (dx)∫2 (dy); ∫1 §(∫2 §∫3 ) =
(∫1 § ∫2 ) § ∫3 . 7 Chapter 26: Change of variable in the integral
Introduction
261 Vitali’s theorem in R r 287
287 262 Lipschitz and di↵erentiable functions 295 Vitali’s theorem for balls in R r ; Lebesgue’s Density Theorem.
Lipschitz functions; elementary properties; di↵erentiable functions from R r to R s ; di↵erentiability and
partial derivatives; approximating a di↵erentiable function by piecewise affine functions; *Rademacher’s
theorem. 263 Di↵erentiableR transformations
in R r
R 308 In the formula g(y)dy = J(x)g(¡(x))dx, find J when ¡ is (i) linear (ii) di↵erentiable; detailed conditions of applicability; polar coordinates; the one-dimensional case. 264 Hausdor↵ measures 319 r-dimensional Hausdor↵ measure on R s ; Borel sets are measurable; Lipschitz functions; if s = r, we have
a multiple of Lebesgue measure; *Cantor measure as a Hausdor↵ measure. 265 Surface measures 329 Normalized Hausdor↵ measure; action of linear operators and di↵erentiable functions; surface measure
on a sphere. Chapter 27: Probability theory
Introduction
271 Distributions 338
339 Terminology; distributions as Radon measures; distribution functions; densities; transformations of random variables. 272 Independence 346 Independent families of random variables; characterizations of independence; joint distributions of (finite)
independent families, and product measures; the zero-one law; E(X£Y ), Var(X + Y ); distribution of a
sum as convolution of distributions; *Etemadi’s inequality. 273 The strong law of large numbers P1
1
Xi !0 a.e. if the Xn are independent with zero expectation and either (i)
n=0 (n+1)2
P1
Var(Xn ) < 1 or (ii) n=0 E(|Xn |1+± ) < 1 for some ± > 0 or (iii) the Xn are identically distributed.
1
n+1 Pn 357 i=0 274 The Central Limit Theorem Normally distributed r.v.s; Lindeberg’s conditions for the Central Limit Theorem; corollaries; estimating
R 1 °x2 /2
dx.
Æ e 275 Martingales 370
381 Sequences of æ-algebras, and martingales adapted to them; up-crossings; Doob’s Martingale Convergence
Theorem; uniform integrability, k k1 -convergence and martingales as sequences of conditional expectations; reverse martingales; stopping times. 276 Martingale di↵erence sequences 392 Martingale di↵erence sequences; strong law of large numbers for m.d.ss.; Koml´
os’ theorem. Chapter 28: Fourier analysis
Introduction
281 The Stone-Weierstrass theorem 401
401 Approximating a function on a compact set by members of a given lattice or algebra of functions; real
and complex cases; approximation by polynomials and trigonometric functions; Weyl’s Equidistribution
Theorem in [0, 1]r . 282 Fourier series 413 Fourier and F´
ejer sums; Dirichlet and F´
ejer kernels; Riemann-Lebesgue lemma; uniform convergence
of F´
ejer sums of a continuous function; a.e. convergence of F´
ejer sums of an integrable function; k k2 convergence of Fourier sums of a square-integrable function; convergence of Fourier sums of a di↵erentiable or b.v. function; convolutions and Fourier coefficients. 283 Fourier Transforms I R
Fourier and inverse Fourier transforms; elementary properties; 01 x1 sin x dx =
R
R
^
2
^
for di↵erentiable and b.v. f ; convolutions; e°x /2 ; f £ g = f £g. 1
º;
2 ^_ the formula f 432
=f 8 284 Fourier Transforms II
^_ 448 Test functions; h = h; tempered functions; tempered functions which represent each other’s transforms;
convolutions; square-integrable functions; Dirac’s delta function. 285 Characteristic functions
The characteristic function of a distribution; independent r.v.s; the normal distribution; the vague
topology on the space of distributions, and sequential convergence of characteristic functions; Poisson’s
theorem. 286 Carleson’s theorem 465 480 The Hardy-Littlewood Maximal Theorem; the Lacey-Thiele proof of Carleson’s theorem. Appendix to Volume 2
Introduction
2A1 Set theory 513
513 Ordered sets; transfinite recursion; ordinals; initial ordinals; Schr¨
oder-Bernstein theorem; filters; Axiom
of Choice; Zermelo’s Well-Ordering Theorem; Zorn’s Lemma; ultrafilters. 2A2 The topology of Euclidean space
Closures; compact sets; open sets in R. 2A3 General topology Topologies; continuous functions; subspace topologies; Hausdor↵ topologies; pseudometrics; convergence
of sequences; compact spaces; cluster points of sequences; convergence of filters; product topologies;
dense subsets. 2A4 Normed spaces 519
523 532 Normed spaces; linear subspaces; Banach spaces; bounded linear operators; dual spaces; extending a
linear operator from a dense subspace; normed algebras. 2A5 Linear topological spaces 535 Linear topological spaces; topologies defined by functionals; completeness; weak topologies. 2A6 Factorization of matrices 539 Determinants; orthonormal families; T = P DQ where D is diagonal and P , Q are orthogonal. Concordance 541 References for Volume 2 542 Index to Volumes 1 and 2
Principal topics and results
General index 544
548 12 Taxonomy of measure spaces *Chapter 21
Taxonomy of measure spaces
I begin this volume with a ‘starred chapter’. The point is that I do not really recommend this chapter
for beginners. It deals with a variety of technical questions which are of great importance for the later
development of the subject, but are likely to be both abstract and obscure for anyone who has not encountered the problems these techniques are designed to solve. On the other hand, if (as is customary) this
work is omitted, and the ideas are introduced only when urgently needed, the student is likely to finish with
very vague ideas on which theorems can be expected to apply to which types of measure space, and with
no vocabulary in which to express those ideas. I therefore take a few pages to introduce the terminology
and concepts which can be used to distinguish ‘good’ measure spaces from others, with a few of the basic
relationships. The only paragraphs which are immediately relevant to the theory set out in Volume 1 are
those on ‘complete’, ‘æ-finite’ and ‘semi-finite’ measure spaces (211A, 211D, 211F, 211Lc, §212, 213A-213B,
215B), and on Lebesgue measure (211M). For the rest, I think that a newcomer to the subject can very
well pass over this chapter for the time being, and return to it for particular items when the text of later
chapters refers to it. On the other hand, it can also be used as an introduction to the flavour of the ‘purest’
kind of measure theory, the study of measure spaces for their own sake, with a systematic discussion of a
few of the elementary constructions. 211 Definitions
I start with a list of definitions, corresponding to the concepts which I have found to be of value in
distinguishing di↵erent types of measure space. Necessarily, the significance of many of these ideas is likely
to be obscure until you have encountered some of the obstacles which arise later on. Nevertheless, you
will I hope be able to deal with these definitions on a formal, abstract basis, and to follow the elementary
arguments involved in establishing the relationships between them (211L).
In §216 I give three substantial examples to demonstrate the rich variety of objects which the definition of
‘measure space’ encompasses. In the present section, therefore, I content myself with very brief descriptions
of sufficient cases to show at least that each of the definitions here discriminates between di↵erent spaces
(211M-211R).
211A Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is (Carath´
eodory) complete
if whenever A µ E 2 ⌃ and µE = 0 then A 2 ⌃; that is, if every negligible subset of X is measurable.
211B Definition Let (X, ⌃, µ) be a measure space. Then (X, ⌃, µ), is a probability space if µX = 1.
In this case µ is called a probability or probability measure.
211C Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is totally finite if µX < 1.
211D Definition Let (X, ⌃, µ) be a measure space. Then µ, or
S (X, ⌃, µ), is æ-finite if there is a
sequence hEn in2N of measurable sets of finite measure such that X = n2N En .
Remark Note that in this case we can set Fn = En \ S i<n Ei , Gn = S i∑n Ei for each n, to obtain a disjoint cover hFn in2N of X by measurable sets of finite measure, and a non-decreasing
sequence hGn in2N of sets of finite measure covering X.
211E Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is strictly localizable or
decomposable
if there is a disjoint family hXi ii2I of measurable sets of finite measure such that X =
S
i2I Xi and
⌃ = {E : E µ X, E \ Xi 2 ⌃ 8 i 2 I}, 211L Definitions µE = P i2I 13 µ(E \ Xi ) for every E 2 ⌃. I will call such a family hXi ii2I a decomposition of X.
P
Remark In this context, we can interpret the sum i2I µ(E \ Xi ) simply as
P
sup{ i2J µ(E \ Xi ) : J is a finite subset of I},
P
taking i2; µ(E \ Xi ) = 0, because we are concerned only with sums of non-negative terms (cf. 112Bd). 211F Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is semi-finite if whenever
E 2 ⌃ and µE = 1 there is an F µ E such that F 2 ⌃ and 0 < µF < 1.
211G Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is localizable or Maharam
if it is semi-finite and, for every E µ ⌃, there is an H 2 ⌃ such that (i) E \ H is negligible for every E 2 E
(ii) if G 2 ⌃ and E \ G is negligible for every E 2 E, then H \ G is negligible. It will be convenient to call
such a set H an essential supremum of E in ⌃. Remark The definition here is clumsy, because really the concept applies to measure algebras rather than to
measure spaces (see 211Yb-211Yc). However, the present definition can be made to work (see 213N, 241G,
243G below) and enables us to proceed without a formal introduction to the concept of ‘measure algebra’
before the time comes to do the job properly in Volume 3.
211H Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is locally determined if it
is semi-finite and
⌃ = {E : E µ X, E \ F 2 ⌃ whenever F 2 ⌃ and µF < 1}; that is to say, for any E 2 PX \ ⌃ there is an F 2 ⌃ such that µF < 1 and E \ F 2
/ ⌃.
211I Definition Let (X, ⌃, µ) be a measure space. A set E 2 ⌃ is an atom for µ if µE > 0 and
whenever F 2 ⌃, F µ E one of F , E \ F is negligible.
211J Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is atomless or di↵used if
there are no atoms for µ. (Note that this is not the same thing as saying that all finite sets are negligible;
see 211R below.)
211K Definition Let (X, ⌃, µ) be a measure space. Then µ, or (X, ⌃, µ), is purely atomic if whenever
E 2 ⌃ and E is not negligible there is an atom for µ included in E.
Remark
P Recall that a measure µ on a set X is point-supported if µ measures every subset of X and
µE = x2E µ{x} for every E µ X (112Bd). Every point-supported measure is purely atomic, because {x}
must be an atom whenever µ{x} > 0, but not every purely atomic measure is point-supported (211R).
211L The relationships between the concepts above are in a sense very straightforward; all the direct
implications in which one property implies another are given in the next theorem.
Theorem (a) A probability space is totally finite.
(b) A totally finite measure space is æ-finite.
(c) A æ-finite measure space is strictly localizable.
(d) A strictly localizable measure space is localizable and locally determined.
(e) A localizable measure space is semi-finite.
(f) A locally determined measure space is semi-finite.
proof (a), (b), (e) and (f) are trivial.
(c) Let (X, ⌃, µ) be a æ-finite measure space; let hFn in2N be a disjoint sequence of measurable sets of
finite measure covering X (see the remark in 211D). If E 2 ⌃, then of course E \ Fn 2 ⌃ for every n 2 N,
and 14 Taxonomy of measure spaces µE = P1 n=0 µ(E \ Fn ) = P n2N 211L µ(E \ Fn ). If E µ X and E \ Fn 2 ⌃ for every n 2 N, then
S
E = n2N E \ Fn 2 ⌃. So hFn in2N is a decomposition of X and (X, ⌃, µ) is strictly localizable. (d) Let (X, ⌃, µ) be a strictly localizable measure space; let hXi ii2I be a decomposition of X. (i) Let E be a family of measurable subsets of X. Let F be the family of measurable sets F µ X
such
that µ(F \ E) = 0 for every E 2 E. Note that ; 2 F and, if hFn in2N is any sequence in F, then
S
n2N Fn 2 F. For each i 2 I, set ∞i = sup{µ(F \ Xi ) : F 2 F} and choose a sequence hFin in2N in F such
that limn!1 µ(Fin \ Xi ) = ∞i ; set
S
Fi = n2N Fin 2 F .
Set F = S i2I F i \ Xi µ X and H = X \ F .
We see that F \ Xi = Fi \ Xi for each i 2 I (because hXi ii2I is disjoint), so F 2 ⌃ and H 2 ⌃. For any
E 2 E,
P
P
µ(E \ H) = µ(E \ F ) = i2I µ(E \ F \ Xi ) = i2I µ(E \ Fi \ Xi ) = 0
because every Fi belongs to F. Thus F 2 F. If G 2 ⌃ and µ(E \ G) = 0 for every E 2 E, then X \ G and
F 0 = F [(X\G) belong to F. So µ(F 0 \Xi ) ∑ ∞i for each i 2 I. But also µ(F \Xi ) ∏ supn2N µ(Fin \Xi ) = ∞i ,
so µ(F \ Xi ) = µ(F 0 \ Xi ) for each i. Because µXi is finite, it follows that µ((F 0 \ F ) \ Xi ) = 0, for each i.
Summing over i, µ(F 0 \ F ) = 0, that is, µ(H \ G) = 0.
Thus H is an essential supremum f...

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