MEASURE THEORY Volume 2 - NTNU · PDF fileTopological Riesz Spaces and Measure Theory,...

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MEASURE THEORY Volume 2 D.H.Fremlin

Transcript of MEASURE THEORY Volume 2 - NTNU · PDF fileTopological Riesz Spaces and Measure Theory,...

  • MEASURE THEORY

    Volume 2

    D.H.Fremlin

  • By the same author:Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.Consequences of Martins Axiom, Cambridge University Press, 1982.

    Companions to the present volume:Measure Theory, vol. 1, Torres Fremlin, 2000;Measure Theory, vol. 3, Torres Fremlin, 2002;Measure Theory, vol. 4, Torres Fremlin, 2003;Measure Theore, vol. 5, Torres Fremlin, 2008.

    First edition May 2001

    Second edition January 2010

  • MEASURE THEORY

    Volume 2

    Broad Foundations

    D.H.Fremlin

    Research Professor in Mathematics, University of Essex

  • Dedicated by the Author

    to the Publisher

    This book may be ordered from the printers, http://www.lulu.com/buy.

    First published in 2001by Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, England

    c D.H.Fremlin 2001The 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 inhttp://www.gnu.org/licenses/dsl.html. For the source files see http://www.essex.ac.uk/maths/staff/fremlin/mt2.2010/index.htm.

    Library of Congress classification QA312.F72

    AMS 2010 classification 28-01

    ISBN 978-0-9538129-7-4

    Typeset by AMS-TEXPrinted by Lulu.com

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    Contents

    General Introduction 9

    Introduction to Volume 2 10

    *Chapter 21: Taxonomy of measure spaces

    Introduction 12211 Definitions 12

    Complete, totally finite, -finite, strictly localizable, semi-finite, localizable, locally determined measure spaces; atoms;

    elementary relationships; countable-cocountable measures.

    212 Complete spaces 17Measurable and integrable functions on complete spaces; completion of a measure.

    213 Semi-finite, locally determined and localizable spaces 23Integration on semi-finite spaces; c.l.d. versions; measurable envelopes; characterizing localizability, strict localizabil-ity, -finiteness.

    214 Subspaces 33Subspace measures on arbitrary subsets; integration; direct sums of measure spaces; *extending measures to well-

    ordered families of sets.

    215 -finite spaces and the principle of exhaustion 43The principle of exhaustion; characterizations of -finiteness; the intermediate value theorem for atomless measures.

    *216 Examples 46A complete localizable non-locally-determined space; a complete locally determined non-localizable space; a completelocally determined localizable space which is not strictly localizable.

    Chapter 22: The Fundamental Theorem of Calculus

    Introduction 52221 Vitalis theorem in R 52

    Vitalis theorem for intervals in R.

    222 Differentiating an indefinite integral 55Monotonic functions are differentiable a.e., and their derivatives are integrable; d

    dx

    xa f = f a.e.; *the Denjoy-Young-

    Saks theorem.

    223 Lebesgues density theorems 63f(x) = limh0

    12h

    x+hxh f a.e. (x); density points; limh0

    12h

    x+hxh |f f(x)| = 0 a.e. (x); the Lebesgue set of a

    function.

    224 Functions of bounded variation 67Variation of a function; differences of monotonic functions; sums and products, limits, continuity and differentiabilityfor b.v. functions; an inequality for

    fg.

    225 Absolutely continuous functions 75Absolute 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 84Sums over arbitrary index sets; saltus functions; the Lebesgue decomposition.

    Chapter 23: The Radon-Nikodym theorem

    Introduction 92231 Countably additive functionals 92

    Additive and countably additive functionals; Jordan and Hahn decompositions.

    232 The Radon-Nikodym theorem 96Absolutely and truly continuous additive functionals; truly continuous functionals are indefinite integrals; *theLebesgue decomposition of a countably additive functional.

    233 Conditional expectations 105-subalgebras; conditional expectations of integrable functions; convex functions; Jensens inequality.

    234 Operations on measures 112Inverse-measure-preserving functions; image measures; sums of measures; indefinite-integral measures; ordering ofmeasures.

    235 Measurable transformations 122The formula

    g(y)(dy) =

    J(x)g((x))(dx); detailed conditions of applicability; inverse-measure-preserving func-

    tions; the image measure catastrophe; using the Radon-Nikodym theorem.

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    Chapter 24: Function spaces

    Introduction 133241 L0 and L0 133

    The linear, order and multiplicative structure of L0; Dedekind completeness and localizability; action of Borel func-tions.

    242 L1 141The normed lattice L1; integration as a linear functional; completeness and Dedekind completeness; the Radon-

    Nikodym theorem and conditional expectations; convex functions; dense subspaces.

    243 L 151The normed lattice L; norm-completeness; the duality between L1 and L; localizability, Dedekind completenessand the identification L = (L1).

    244 Lp 158The normed lattices Lp, for 1 < p < ; Holders inequality; completeness and Dedekind completeness; (Lp) = Lq ;conditional expectations; *uniform convexity.

    245 Convergence in measure 172The topology of (local) convergence in measure on L0; pointwise convergence; localizability and Dedekind complete-ness; embedding Lp in L0; 1-convergence and convergence in measure; -finite spaces, metrizability and sequentialconvergence.

    246 Uniform integrability 183Uniformly integrable sets in L1 and L1; elementary properties; disjoint-sequence characterizations; 1 and conver-gence in measure on uniformly integrable sets.

    247 Weak compactness in L1 191A subset of L1 is uniformly integrable iff it is relatively weakly compact.

    Chapter 25: Product measures

    Introduction 196251 Finite products 196

    Primitive and c.l.d. products; basic properties; Lebesgue measure on Rr+s as a product measure; products of directsums and subspaces; c.l.d. versions.

    252 Fubinis theorem 212When

    f(x, y)dxdy and

    f(x, y)d(x, y) are equal; measures of ordinate sets; *the volume of a ball in Rr.

    253 Tensor products 228Bilinear operators; bilinear operators L1()L1()W and linear operators L1()W ; positive bilinear operatorsand the ordering of L1( ); conditional expectations; upper integrals.

    254 Infinite products 238Products 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 determinedby 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 tosubproducts.

    255 Convolutions of functions 255Shifts in R2 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; f g1 f1g1; the groups Rr and ], ].256 Radon measures on Rr 267

    Definition of Radon measures on Rr; completions of Borel measures; Lusin measurability; image measures; productsof two Radon measures; semi-continuous functions.

    257 Convolutions of measures 274Convolution of totally finite Radon measures on Rr;

    h d(12) =

    h(x+y)1(dx)2(dy); 1(23) = (12)3.

    Chapter 26: Change of variable in the integral

    Introduction 277261 Vitalis theorem in Rr 277

    Vitalis theorem for balls in Rr; Lebesgues Density Theorem.

    262 Lipschitz and differentiable functions 285Lipschitz functions; elementary properties; differentiable functions from Rr to Rs; differentiability and partial deriva-tives; approximating a differentiable function by piecewise affine functions; *Rademachers theorem.

    263 Differentiable transformations in Rr 296In the formula

    g(y)dy =

    J(x)g((x))dx, find J when is (i) linear (ii) differentiable; detailed conditions of

    applicability; polar coordinates; the one-dimensional case.

    264 Hausdorff measures 307r-dimensional Hausdorff measure on Rs; Borel sets are measurable; Lipschitz functions; if s = r, we have a multiple

    of Lebesgue measure; *Cantor measure as a Hausdorff measure.

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    265 Surface measures 316Normalized Hausdorff measure; action of linear operators and differentiable functions; surface measure on a sphere.

    *266 The Brunn-Minkowski inequality 324Arithmetic and geometric means; essential closures; the Brunn-Minkowski inequality.

    Chapter 27: Probability theory

    Introduction 328271 Distributions 329

    Terminology; distributions as Radon measures; distribution functions; densities; transformations of random variables;

    *distribution functions and convergence in measure.

    272 Independence 335Independent families of random variables; characterizations of independence; joint distributions of (finite) independent

    families, and product measures; the zero-one law; E(XY ), Var(X + Y ); distribution of a sum as convolution ofdistributions; Etemadis inequality; *Hoeffdings inequality.

    273 The strong law of large numbers 3481

    n+1

    ni=0 Xi0 a.e. if the Xn are independent with zero expectation and either (i)

    n=0

    1(n+1)2

    Var(Xn) < or(ii)

    n=0 E(|Xn|1+) < for some > 0 or (iii) the Xn are identically distributed.

    274 The Central Limit Theorem 360Normally distributed r.v.s; Lindebergs conditions for the Central Limit Theorem; corollaries; estimating

    e