MEASURE THEORYVolume 4D.H.FremlinBy 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. 2, Torres Fremlin, 2001.Measure Theory, vol. 3, Torres Fremlin, 2002.First printing November 2003MEASURE THEORYVolume 4Topological Measure SpacesD.H.FremlinResearch Professor in Mathematics, University of EssexDedicated by the Authorto the PublisherThis book may be ordered from the publisher at the address below. For price and means of pay-ment see the authors Web page http://www.essex.ac.uk/maths/staff/fremlin/mtsales.htm,or enquire from fremdh@essex.ac.uk.First published in 2003by Torres Fremlin, 25 Ireton Road, Colchester CO3 3AT, Englandc _ D.H.Fremlin 2003The right of D.H.Fremlin to be identied as author of this work has been asserted in accordance withthe Copyright, Designs and Patents Act 1988. This work is issued under the terms of the Design ScienceLicense as published in http://dsl.org/copyleft/dsl.txt. For the source les see http://www.essex.ac.uk/maths/staff/fremlin/mt4.2003/index.htm.Library of Congress classication QA312.F72AMS 2000 classication 28A99ISBN 0-9538129-4-4Typeset by //o-TEXPrinted in England by Biddles Short Run Books, Kings Lynn5ContentsGeneral Introduction 10Introduction to Volume 4 11Chapter 41: Topologies and measures IIntroduction 13411 Denitions 13Topological, inner regular, -additive, outer regular, locally nite, eectively locally nite, quasi-Radon,Radon, completion regular, Baire, Borel and strictly positive measures; measurable and almost contin-uous functions; self-supporting sets and supports of measures; Stone spaces; Dieudonnes measure.412 Inner regularity 19Exhaustion; Baire measures; Borel measures on metrizable spaces; completions and c.l.d. versions; com-plete locally determined spaces; inverse-measure-preserving functions; subspaces; indenite-integral mea-sures; products; outer regularity.413 Inner measure constructions 31Inner measures; constructing a measure from an inner measure; the inner measure dened by a measure;complete locally determined spaces; extension of functionals to measures; countably compact classes;constructing measures dominating given functionals.414 -additivity 50Semi-continuous functions; supports; strict localizability; subspace measures; regular topologies; densitytopologies; lifting topologies.415 Quasi-Radon measure spaces 58Strict localizability; subspaces; regular topologies; hereditarily Lindelof spaces; products of separablemetrizable spaces; comparison and specication of quasi-Radon measures; construction of quasi-Radonmeasures extending given functionals; indenite-integral measures; Lpspaces; Stone spaces.416 Radon measure spaces 73Radon and quasi-Radon measures; specication of Radon measures; c.l.d. versions of Borel measures;locally compact topologies; constructions of Radon measures extending or dominating given functionals;additive functionals on Boolean algebras and Radon measures on Stone spaces; subspaces; products;Stone spaces of measure algebras; compact and perfect measures; representation of homomorphisms ofmeasure algebras; the split interval.417 -additive product measures 88The product of two eectively locally nite -additive measures; the product of many -additive proba-bility measures; Fubinis theorem; generalized associative law; measures on subproducts as image mea-sures; products of strictly positive measures; quasi-Radon and Radon product measures; when ordinaryproduct measures are -additive; continuous functions and Baire -algebras in product spaces.418 Measurable functions and almost continuous functions 110Measurable functions; into (separable) metrizable spaces; and image measures; almost continuous func-tions; continuity, measurability, image measures; expressing Radon measures as images of Radon mea-sures; Prokhorovs theorem on projective limits of Radon measures; representing measurable functionsinto L0spaces.419 Examples 126A nearly quasi-Radon measure; a Radon measure space in which the Borel sets are inadequate; a nearlyRadon measure; the Stone space of the Lebesgue measure algebra; measures with domain 11; notes onLebesgue measure.Chapter 42: Descriptive set theoryIntroduction 138421 Souslins operation 138Souslins operation; is idempotent; as a projection operator; Souslin-F sets; *constituents.422 K-analytic spaces 148Usco-compact relations; K-analytic sets; and Souslin-F sets; *First Separation Theorem.423 Analytic spaces 155Analytic spaces; are K-analytic spaces with countable networks; Souslin-F sets; Borel measurable func-tions; injective images of Polish spaces; non-Borel analytic sets; von Neumann-Jankow selection theorem;*constituents of coanalytic sets.424 Standard Borel spaces 165Elementary properties; isomorphism types; subspaces; Borel measurable actions of Polish groups.6Chapter 43: Topologies and measures IIIntroduction 172431 Souslins operation 172The domain of a complete locally determined measure is closed under Souslins operation; the kernel ofa Souslin scheme is approximable from within.432 K-analytic spaces 176Topological measures on K-analytic spaces; extensions to Radon measures; expressing Radon measuresas images of Radon measures.433 Analytic spaces 180Measures on spaces with countable networks; inner regularity of Borel measures; expressing Radonmeasures as images of Radon measures; measurable and almost continuous functions; the von Neumann-Jankow selection theorem; products; extension of measures on -subalgebras; standard Borel spaces.434 Borel measures 184Classication of Borel measures; Radon spaces; universally measurable sets and functions; Borel-measure-compact, Borel-measure-complete and pre-Radon spaces; countable compactness and countable tight-ness; quasi-dyadic spaces and completion regular measures; rst-countable spaces and Borel productmeasures.435 Baire measures 203Classication of Baire measures; extension of Baire measures to Borel measures (Mariks theorem);measure-compact spaces; sequential spaces and Baire product measures.436 Representation of linear functionals 209Smooth and sequentially smooth linear functionals; measures and sequentially smooth functionals; Bairemeasures; products of Baire measures; quasi-Radon measures and smooth functionals; locally compactspaces and Radon measures.437 Spaces of measures 219Smooth and sequentially smooth duals; signed measures; embedding spaces of measurable functions inthe bidual of Cb(X); vague and narrow topologies; product measures; extreme points; uniform tightness;Prokhorov spaces.438 Measure-free cardinals 240Measure-free cardinals; point-nite families of sets with measurable unions; measurable functions intometrizable spaces; Radon and measure-compact metric spaces; metacompact spaces; hereditarily weakly-renable spaces; when c is measure-free.439 Examples 254Measures with no extensions to Borel measures; universally negligible sets; Hausdor measures are rarelysemi-nite; a smooth linear functional not expressible as an integral; a rst-countable non-Radon space;Baire measures not extending to Borel measures; Ncis not Borel-measure-compact; the Sorgenfrey line;Q is not a Prokhorov space.Chapter 44: Topological groupsIntroduction 271441 Invariant measures on locally compact spaces 271Measures invariant under homeomorphisms; Haar measures; measures invariant under isometries.442 Uniqueness of Haar measure 280Two (left) Haar measures are multiples of each other; left and right Haar measures; Haar measurableand Haar negligible sets; the modular function of a group; formulae for_f(x1)dx, _f(xy)dx.443 Further properties of Haar measure 287The Haar measure algebra of a group carrying Haar measures; actions of the group on the Haar measurealgebra; locally compact groups; actions of the group on L0and Lp; the bilateral uniformity; Borel setsare adequate; completing the group; expressing an arbitrary Haar measure in terms of a Haar measureon a locally compact group; completion regularity of Haar measure; invariant measures on the set of leftcosets of a closed subgroup of a locally compact group; modular functions of subgroups and quotientgroups; transitive actions of compact groups on compact spaces.444 Convolutions 311Convolutions of quasi-Radon measures; the Banach algebra of signed -additive measures; continuousactions and corresponding actions on L0() for an arbitrary quasi-Radon measure ; convolutions ofmeasures and functions; indenite-integral measures over a Haar measure ; convolutions of functions;Lp(); approximate identities.7445 The duality theorem 334Dual groups; Fourier-Stieltjes transforms; Fourier transforms; identifying the dual group with the max-imal ideal space of L1; the topology of the dual group; positive denite functions; Bochners theorem;the Inversion Theorem; the Plancherel Theorem; the Duality Theorem.446 The structure of locally compact groups 357Finite-dimensional representations separate the points of a compact group; groups with no small sub-groups have B-sequences; chains of subgroups.447 Translation-invariant liftings 372Translation-invariant liftings and lower densities; Vitalis theorem and a density theorem for groups withB-sequences; Haar measures have translation-invariant liftings.448 Invariant measures on Polish spaces 383Countably full local semigroups of Aut A; -equidecomposability; countably non-paradoxical groups;G-invariant additive functions from A to L(C); measures invariant under Polish group actions (theNadkarni-Becker-Kechris theorem).449 Amenable groups 393Amenable groups; permanence properties; locally compact amenable groups; Tarskis theorem; discreteamenable groups.Chapter 45: Perfect measures, disintegrations and processesIntroduction 413451 Perfect, compact and countably compact measures 414Basic properties of the three classes; subspaces, completions, c.l.d. versions, products; measurable func-tions from compact measure spaces to metrizable spaces; *weakly -favourable spaces.452 Integration and disintegration of measures 427Integrating families of probability measures; -additive and Radon measures; disintegrations and regularconditional probabilities; disintegrating countably compact measures; disintegrating Radon measures;*images of countably compact measures.453 Strong liftings 441Strong and almost strong liftings; existence; on product spaces; disintegrations of Radon measures overspaces with almost strong liftings; Stone spaces; Loserts example.454 Measures on product spaces 454Perfect, compact and countably compact measures on product spaces; extension of nitely additivefunctions with perfect countably additive marginals; Kolmogorovs extension theorem; measures denedfrom conditional distributions; distributions of random processes; measures on C(T) for Polish T.455 Markov process and Brownian motion 464Denition of Markov process from conditional distributions; existence of a measure representing Brown-ian motion; continuous sample paths.456 Gaussian distributions 472Gaussian distributions; supports; universal Gaussian distributions; cluster sets of n-dimensional pro-cesses; -additivity; Gaussian processes.457 Simultaneous extension of measures 488Extending families of nitely additive functionals; Strassens theorem; extending families of measures;examples.458 Relative independence and relative products 497Relatively independent families of -algebras and random variables; relative distributions; relativelyindependent families of closed subalgebras of a probability algebra; relative free products of probabilityalgebras; relative products of probability spaces; existence of relative products.459 Symmetric measures and exchangeable random variables 510Exchangeable families of inverse-measure-preserving functions; de Finettis theorem; countably compactsymmetric measures on product spaces disintegrate into product measures; symmetric quasi-Radonmeasures.Chapter 46: Pointwise compact sets of measurable functionsIntroduction 522461 Barycenters and Choquets theorem 522Barycenters; elementary properties; sucient conditions for existence; closed convex hulls of compactsets; Krens theorem; measures on sets of extreme points.462 Pointwise compact sets of continuous functions 532Angelic spaces; the topology of pointwise convergence on C(X); weak convergence and weakly compactsets in C0(X); Radon measures on C(X); separately continuous functions; convex hulls.8463 Tp and Tm 538Pointwise convergence and convergence in measure on spaces of measurable functions; compact andsequentially compact sets; perfect measures and Fremlins Alternative; separately continuous functions.464 Talagrands measure 550The usual measure on 1I; the intersection of a sequence of non-measurable lters; Talagrands measure;the L-space of additive functionals on 1I; measurable and purely non-measurable functionals.465 Stable sets 563Stable sets of functions; elementary properties; pointwise compactness; pointwise convergence and con-vergence in measure; a law of large numbers; stable sets and uniform convergence in the strong law oflarge numbers; stable sets in L0and L1; *R-stable sets.466 Measures on linear topological spaces 590Quasi-Radon measures for weak and strong topologies; Kadec norms; constructing weak-Borel measures;characteristic functions of measures on locally convex spaces; universally measurable linear operators.*467 Locally uniformly rotund norms 598Locally uniformly rotund norms; separable normed spaces; long sequences of projections; K-countablydetermined spaces; weakly compactly generated spaces; Banach lattices with order-continuous norms;Eberlein compacta.Chapter 47: Geometric measure theoryIntroduction 610471 Hausdor measures 610Metric outer measures; Increasing Sets Lemma; analytic spaces; inner regularity; Vitalis theorem and adensity theorem; Howroyds theorem.472 Besicovitchs Density Theorem 626Besicovitchs Covering Lemma; Besicovitchs Density Theorem; *a maximal theorem.473 Poincares inequality 633Dierentiable and Lipschitz functions; smoothing by convolution; the Gagliardo-Nirenberg-Sobolev in-equality; Poincares inequality for balls.474 The distributional perimeter 647The divergence of a vector eld; sets with locally nite perimeter, perimeter measures and outward-normal functions; the reduced boundary; invariance under isometries; isoperimetric inequalities; Federerexterior normals; the Compactness Theorem.475 The essential boundary 670Essential interior, closure and boundary; the reduced boundary; perimeter measures; characterizing setswith locally nite perimeter; the Divergence Theorem; calculating perimeters from cross-sectional counts;Cauchys Perimeter Theorem; the Isoperimetric Theorem for convex sets.476 Concentration of measure 690Hausdor metrics; Vietoris topologies; concentration by partial reection; concentration of measure inRr; the Isoperimetric Theorem; concentration of measure on spheres.Chapter 48: Gauge integralsIntroduction 704481 Tagged partitions 704Tagged partitions and Riemann sums; gauge integrals; gauges; residual sets; subdivisions; examples(the Riemann integral, the Henstock integral, the symmetric Riemann-complete integral, the McShaneintegral, box products, the approximately continuous Henstock integral).482 General theory 714Saks-Henstock lemma; when gauge-integrable functions are measurable; when integrable functions aregauge-integrable; I(f H); integrating derivatives; B.Levis theorem; Fubinis theorem.483 The Henstock integral 729The Henstock and Lebesgue integrals; indenite Henstock integrals; Saks-Henstock lemma; fundamentaltheorem of calculus; the Perron integral; ACG functions.484 The Pfeer integral 746The Tamanini-Giacomelli theorem; a family of tagged-partition structures; the Pfeer integral; the Saks-Henstock indenite integral of a Pfeer integrable function; Pfeers Divergence Theorem; dierentiatingthe indenite integral; invariance under lipeomorphisms.9Chapter 49: Further topicsIntroduction 765491 Equidistributed sequences 765The asymptotic density ideal ?; equidistributed sequences; when equidistributed sequences exist; Z =1N/?; eectively regular measures; equidistributed sequences and induced embeddings of measure al-gebras in Z.492 Combinatorial concentration of measure 782Concentration of measure in product spaces; concentration of measure in permutation groups.493 Extremely amenable groups 789Extremely amenable groups; concentrating additive functionals; measure algebras under .; L0; au-tomorphism groups of measure algebras; isometry groups of spheres in inner product spaces; locallycompact groups.494 Cubes in product spaces 800Subsets of measure algebras with non-zero inma; product sets included in given sets of positive measure.495 Poisson point processes 802Poisson distributions; Poisson point processes; disintegrations; transforming disjointness into stochasticindependence; representing Poisson point processes by Radon measures; exponential distributions andPoisson point processes on [0, [.Appendix to Volume 4Introduction 8234A1 Set theory 823Cardinals; closed conal sets and stationary sets; -system lemma; free sets; Ramseys theorem; theMarriage Lemma; lters; normal ultralters; Ostaszewskis ; cardinals of -algebras.4A2 General topology 827Glossary; general constructions; F, G, zero and cozero sets; countable chain condition; separationaxioms; compact and locally compact spaces; Lindelof spaces; Stone-Cech compactications; uniformspaces; rst-countable, sequential, countably tight, metrizable spaces; countable networks; second-countable spaces; separable metrizable spaces; Polish spaces; order topologies.4A3 Topological -algebras 848Borel -algebras; measurable functions; hereditarily Lindelof spaces; second-countable spaces; Polishspaces; 1; Baire -algebras; product spaces; compact spaces; Baire property algebras; cylindrical -algebras.4A4 Locally convex spaces 857Linear topological spaces; locally convex spaces; Hahn-Banach theorem; normed spaces; inner productspaces; max-ow min-cut theorem.4A5 Topological groups 863Group actions; topological groups; uniformities; quotient groups; metrizable groups.4A6 Banach algebras 869Stone-Weierstrass theorem (fourth form); multiplicative linear functionals; spectral radius; invertibleelements; exponentiation.Concordance 873References for Volume 4 874Index to Volumes 1-4Principal topics and results 881General index 89310General introduction In this treatise I aim to give a comprehensive description of modern abstract measuretheory, with some indication of its principal applications. The rst two volumes are set at an introductorylevel; they are intended for students with a solid grounding in the concepts of real analysis, but possibly withrather limited detailed knowledge. As the book proceeds, the level of sophistication and expertise demandedwill increase; thus for the volume on topological measure spaces, familiarity with general topology will beassumed. The emphasis throughout is on the mathematical ideas involved, which in this subject are mostlyto be found in the details of the proofs.My intention is that the book should be usable both as a rst introduction to the subject and as a referencework. For the sake of the rst aim, I try to limit the ideas of the early volumes to those which are reallyessential to the development of the basic theorems. For the sake of the second aim, I try to express these ideasin their full natural generality, and in particular I take care to avoid suggesting any unnecessary restrictionsin their applicability. Of course these principles are to to some extent contradictory. Nevertheless, I nd thatmost of the time they are very nearly reconcilable, provided that I indulge in a certain degree of repetition.For instance, right at the beginning, the puzzle arises: should one develop Lebesgue measure rst on thereal line, and then in spaces of higher dimension, or should one go straight to the multidimensional case? Ibelieve that there is no single correct answer to this question. Most students will nd the one-dimensionalcase easier, and it therefore seems more appropriate for a rst introduction, since even in that case thetechnical problems can be daunting. But certainly every student of measure theory must at a fairly earlystage come to terms with Lebesgue area and volume as well as length; and with the correct formulations, themultidimensional case diers from the one-dimensional case only in a denition and a (substantial) lemma.So what I have done is to write them both out (114-115). In the same spirit, I have been uninhibited,when setting out exercises, by the fact that many of the results I invite students to look for will appear inlater chapters; I believe that throughout mathematics one has a better chance of understanding a theoremif one has previously attempted something similar alone.As I write this Introduction (September 2003), the plan of the work is as follows:Volume 1: The Irreducible MinimumVolume 2: Broad FoundationsVolume 3: Measure AlgebrasVolume 4: Topological Measure SpacesVolume 5: Set-theoretic Measure Theory.Volume 1 is intended for those with no prior knowledge of measure theory, but competent in the elementarytechniques of real analysis. I hope that it will be found useful by undergraduates meeting Lebesgue measurefor the rst time. Volume 2 aims to lay out some of the fundamental results of pure measure theory(the Radon-Nikod ym theorem, Fubinis theorem), but also gives short introductions to some of the mostimportant applications of measure theory (probability theory, Fourier analysis). While I should like tobelieve that most of it is written at a level accessible to anyone who has mastered the contents of Volume 1,I should not myself have the courage to try to cover it in an undergraduate course, though I would certainlyattempt to include some parts of it. Volumes 3 and 4 are set at a rather higher level, suitable to postgraduatecourses; while Volume 5 will assume a wide-ranging competence over large parts of analysis and set theory.There is a disclaimer which I ought to make in a place where you might see it in time to avoid paying forthis book. I make no attempt to describe the history of the subject. This is not because I think the historyuninteresting or unimportant; rather, it is because I have no condence of saying anything which would notbe seriously misleading. Indeed I have very little condence in anything I have ever read concerning thehistory of ideas. So while I am happy to honour the names of Lebesgue and Kolmogorov and Maharam inmore or less appropriate places, and I try to include in the bibliographies the works which I have myselfconsulted, I leave any consideration of the details to those bolder and better qualied than myself.The work as a whole is not yet complete; and when it is nished, it will undoubtedly be too longto be printed as a single volume in any reasonable format. I am therefore publishing it one part at atime. However, drafts of most of the rest are available on the Internet; see http://www.essex.ac.uk/maths/staff/fremlin/mt.htm for detailed instructions. For the time being, at least, printing will be inshort runs. I hope that readers will be energetic in commenting on errors and omissions, since it should bepossible to correct these relatively promptly. An inevitable consequence of this is that paragraph referencesmay go out of date rather quickly. I shall be most attered if anyone chooses to rely on this book as a sourceIntroduction to Volume 4 11for basic material; and I am willing to attempt to maintain a concordance to such references, indicatingwhere migratory results have come to rest for the moment, if authors will supply me with copies of paperswhich use them.I mention some minor points concerning the layout of the material. Most sections conclude with lists ofbasic exercises and further exercises, which I hope will be generally instructive and occasionally enter-taining. How many of these you should attempt must be for you and your teacher, if any, to decide, as notwo students will have quite the same needs. I mark with a >>> those which seem to me to be particularlyimportant. But while you may not need to write out solutions to all the basic exercises, if you are in anydoubt as to your capacity to do so you should take this as a warning to slow down a bit. The furtherexercises are unbounded in diculty, and are unied only by a presumption that each has at least onesolution based on ideas already introduced. Occasionally I add a nal problem, a question to which I donot know the answer and which seems to arise naturally in the course of the work.The impulse to write this book is in large part a desire to present a unied account of the subject.Cross-references are correspondingly abundant and wide-ranging. In order to be able to refer freely acrossthe whole text, I have chosen a reference system which gives the same code name to a paragraph whereverit is being called from. Thus 132E is the fth paragraph in the second section of the third chapter ofVolume 1, and is referred to by that name throughout. Let me emphasize that cross-references are supposedto help the reader, not distract her. Do not take the interpolation (121A) as an instruction, or even arecommendation, to lift Volume 1 o the shelf and hunt for 121. If you are happy with an argument as itstands, independently of the reference, then carry on. If, however, I seem to have made rather a large jump,or the notation has suddenly become opaque, local cross-references may help you to ll in the gaps.Each volume will have an appendix of useful facts, in which I set out material which is called onsomewhere in that volume, and which I do not feel I can take for granted. Typically the arrangement ofmaterial in these appendices is directed very narrowly at the particular applications I have in mind, and isunlikely to be a satisfactory substitute for conventional treatments of the topics touched on. Moreover, theideas may well be needed only on rare and isolated occasions. So as a rule I recommend you to ignore theappendices until you have some direct reason to suppose that a fragment may be useful to you.During the extended gestation of this project I have been helped by many people, and I hope that myfriends and colleagues will be pleased when they recognise their ideas scattered through the pages below.But I am especially grateful to those who have taken the trouble to read through earlier drafts and commenton obscurities and errors.Introduction to Volume 4I return in this volume to the study of measure spaces rather than measure algebras. For fty years nowmeasure theory has been intimately connected with general topology. Not only do a very large proportionof the measure spaces arising in applications carry topologies related in interesting ways to their measures,but many questions in abstract measure theory can be eectively studied by introducing suitable topologies.Consequently any course in measure theory at this level must be frankly dependent on a substantial knowl-edge of topology. With this proviso, I hope that the present volume will be accessible to graduate students,and will lead them to the most important ideas of modern abstract measure theory.The rst and third chapters of the volume seek to provide a thorough introduction into the ways in whichtopologies and measures can interact. They are divided by a short chapter on descriptive set theory, onthe borderline between set theory, logic, real analysis and general topology, which I single out for detailedexposition because I believe that it forms an indispensable part of the background of any measure theorist.Chapter 41 is dominated by the concepts of inner regularity and -additivity, coming together in Radonmeasures (416). Chapter 43 concentrates rather on questions concerning properties of a topological spacewhich force particular relationships with measures on that space. But plenty of side-issues are treated inboth, such as Lusin measurability (418), the denition of measures from linear functionals (436) andmeasure-free cardinals (438). Chapters 45 and 46 continue some of the same themes, with particularinvestigations into disintegrations or regular conditional probabilities (452-453), the abstract theory ofstochastic processes (454-455), Talagrands theory of Glivenko-Cantelli classes (465) and the theory ofmeasures on normed spaces (466-467).In contrast with the relatively amorphous structure of Chapters 41, 43, 45 and 46, four chapters of thisvolume have denite topics. I have already said that Chapter 42 is an introduction to descriptive set theory;12 Introduction to Volume 4like Chapters 31 and 35 in the last volume, it is a kind of appendix brought into the main stream of theargument. Chapter 44 deals with topological groups. Most of it is of course devoted to Haar measure,giving the Pontryagin-van Kampen duality theorem (445) and the Ionescu Tulcea theorem on the existenceof translation-invariant liftings (447). But there are also sections on Polish groups (448) and amenablegroups (449), and some of the general theory of measures on measurable groups (444). Chapter 47 is asecond excursion, after Chapter 26, into geometric measure theory. It starts with Hausdor measures (471),gives a proof of the Di Giorgio-Federer Divergence Theorem (475), and then examines a number of examplesof concentration of measure (476). In Chapter 48, I set out the elementary theory of gauge integrals, withsections on the Henstock and Pfeer integrals (483-484). Finally, in Chapter 49, I give notes on ve specialtopics: equidistributed sequences (491), combinatorial forms of concentration of measure (492), extremelyamenable groups (493), subproducts in product spaces (494) and Poisson point processes (495).I had better mention prerequisites, as usual. To embark on this material you will certainly need a solidfoundation in measure theory. Since I do of course use my own exposition as my principal source of referencesto the elementary ideas, I advise readers to ensure that they have easy access to all three previous volumesbefore starting serious work on this one. But you may not need to read very much of them. It might beprudent to glance through the detailed contents of Volume 1 and the rst ve chapters of Volume 2 to checkthat most of the material there is more or less familiar. But Volume 3, and the last three chapters of Volume2, can probably be left on one side for the moment. Of course you will need the Lifting Theorem (Chapter34) for 447, 452 and 453, and Chapter 26 is essential background for Chapter 47, while Chapter 28 (onFourier analysis) may help to make sense of Chapter 44, and parts of Chapter 27 (on probability theory)are necessary for 455-456. And measure algebras are mentioned in every chapter except (I think) Chapter48; but I hope that the cross-references are precise enough to lead you to just what you need to know atany particular point. Even Maharams theorem is hardly used in this volume.What you will need, apart from any knowledge of measure theory, is a sound background in generaltopology. This volume calls on a great many miscellaneous facts from general topology, and the list in4A2 is not a good place to start if continuity and compactness and the separation axioms are unfamiliar.My primary reference for topology is Engelking 89. I do not insist that you should have read this book(though of course I hope you will do so sometime); but I do think you should make sure that you can use it.In the general introduction to this treatise, I wrote I make no attempt to describe the history of thesubject, and I have generally been casual some would say negligent in my attributions of results totheir discoverers. Through much of the rst three volumes I did at least have the excuse that the historyexists in print in far more detail than I am qualied to describe. In the present volume I nd my positionmore uncomfortable, in that I have been watching the evolution of the subject relatively closely over thelast thirty years, and ought to be able to say something about it. Nevertheless I remain reluctant to makedenite statements crediting one person rather than another with originating an idea. My more intimateknowledge of the topic makes me even more conscious than elsewhere of the danger of error and of thebreadth of reading that would be necessary to produce a balanced account. In some cases I do attach aresult to a specic published paper, but these attributions should never be regarded as an assertion thatany particular author has priority; at most, they declare that a historian should examine the source citedbefore coming to any decision. I assure my friends and colleagues that my omissions are not intended toslight either them or those we all honour. What I have tried to do is to include in the bibliography to thisvolume all the published work which (as far as I am consciously aware) has inuenced me while writing it,so that those who wish to go into the matter will have somewhere to start their investigations.411B Denitions 13Chapter 41Topologies and Measures II begin this volume with an introduction to some of the most important ways in which topologies andmeasures can interact, and with a description of the forms which such constructions as subspaces and productspaces take in such contexts. By far the most important concept is that of Radon measure (411H, 416). InRadon measure spaces we nd both the richest combinations of ideas and the most important applications.But, as usual, we are led both by analysis of these ideas and by other interesting examples to considerwider classes of topological measure space, and the greater part of the chapter, by volume, is taken up by adescription of the many properties of Radon measures individually and in partial combinations.I begin the chapter with a short section of denitions (411), including a handful of more or less elementaryexamples. The two central properties of a Radon measure are inner regularity (411B) and -additivity(411C). The former is an idea of great versatility which I look at in an abstract setting in 412. I take asection (413) to describe some methods of constructing measure spaces, extending the rather limited rangeof constructions oered in earlier volumes. There are two sections on -additive measures, 414 and 417;the former covers the elementary ideas, and the latter looks at product measures, where it turns out thatwe need a new technique to supplement the purely measure-theoretic constructions of Chapter 25. On theway to Radon measures in 416, I pause over quasi-Radon measures (411H, 415), where inner regularityand -additivity rst come eectively together.The possible interactions of a topology and a measure on the same space are so varied that even a briefaccount makes a long chapter; and this is with hardly any mention of results associated with particular typesof topological space, most of which must wait for later chapters. But I include one section on the two mostimportant classes of functions acting between topological measure spaces (418), and another describingsome examples to demonstrate special phenomena (419).411 DenitionsIn something of the spirit of 211, but this time without apologising, I start this volume with a list ofdenitions. The rest of Chapter 41 will be devoted to discussing these denitions and relationships betweenthem, and integrating the new ideas into the concepts and constructions of earlier volumes; I hope that bypresenting the terminology now I can give you a sense of the directions the following sections will take. Iought to remark immediately that there are many cases in which the exact phrasing of the denitions isimportant in ways which may not be immediately apparent.411A I begin with a phrase which will be a useful shorthand for the context in which most, but not all,of the theory here will be developed.Denition A topological measure space is a quadruple (X, T, , ) where (X, , ) is a measure spaceand T is a topology on X such that T , that is, every open set (and therefore every Borel set) ismeasurable.411B Now I come to what are in my view the two most important concepts to master; jointly they willdominate the chapter.Denition Let (X, , ) be a measure space and / a family of sets. I say that is inner regular withrespect to / ifE = supK : K /, K Efor every E . (Cf. 256Ac, 342Aa.)Remark Note that in this denition I do not assume that / , nor even that / TX. But of course will be inner regular with respect to / i it is inner regular with respect to / .It is convenient in this context to interpret sup as 0, so that we have to check the denition only whenE > 0, and need not insist that /.14 Topologies and measures I 411C411C Denition Let (X, , ) be a measure space and T a topology on X. I say that is -additive(the phrase -regular has also been used) if whenever ( is a non-empty upwards-directed family of opensets such that ( and

( then (

() = supG G.Remark Note that in this denition I do not assume that every open set is measurable. Consequently wecannot take it for granted that an extension of a -additive measure will be -additive; on the other hand,the restriction of a -additive measure to any -subalgebra will be -additive.411D Complementary to 411B we have the following.Denition Let (X, , ) be a measure space and H a family of subsets of X. Then is outer regularwith respect to H ifE = infH : H H, H Efor every E .411E I delay discussion of most of the relationships between the concepts here to later in the chapter.But it will be useful to have a basic fact set out immediately.Proposition Let (X, , ) be a measure space and T a topology on X. If is inner regular with respect tothe compact sets, it is -additive.proof Let ( be a non-empty upwards-directed family of measurable open sets such that H =

( . If < H, there is a compact set K H such that K ; now there must be a G ( which includes K,so that G . As is arbitrary, supG G = H.411F In order to deal eciently with measures which are not totally nite, I think we need the followingideas.Denitions Let (X, , ) be a measure space and T a topology on X.(a) I say that is locally nite if every point of X has a neighbourhood of nite measure, that is, ifthe open sets of nite outer measure cover X.(b) I say that is eectively locally nite if for every non-negligible measurable set E X there isa measurable open set G X such that G < and E G is not negligible.Note that an eectively locally nite measure must measure many open sets, while a locally nite measureneed not.(c) This seems a convenient moment at which to introduce the following term. A real-valued function fdened on a subset of X is locally integrable if for every x X there is an open set G containing x suchthat_Gf is dened (in the sense of 214D) and nite.411G Elementary facts (a) If is a locally nite measure on a topological space X, then K < for every compact set K X. PPP The family ( of open sets of nite outer measure is upwards-directed andcovers X, so there must be some G ( including K, in which case K G is nite. QQQ(b) A measure on Rris locally nite i every bounded set has nite outer measure (cf. 256Ab). PPP (i)If every bounded set has nite outer measure then, in particular, every open ball has nite outer measure,so that is locally nite. (ii) If is locally nite and A Rris bounded, then its closure A is compact(2A2F), so that A A is nite, by (a) above. QQQ(c) I should perhaps remark immediately that a locally nite topological measure need not be eectivelylocally nite (419A), and an eectively locally nite measure need not be locally nite (411P).(d) An eectively locally nite measure must be semi-nite.411K Denitions 15(e) A locally nite measure on a Lindelof space X (denition: 4A2A) is -nite. PPP Let ( be the familyof open sets of nite outer measure. Because is locally nite, ( is a cover of X. Because X is Lindelof,there is a sequence Gn)nN in ( covering X. For each n N, there is a measurable set En Gn of nitemeasure, and now En)nN is a sequence of sets of nite measure covering X. QQQ(f ) Let (X, T, , ) be a topological measure space such that is locally nite and inner regular withrespect to the compact sets. Then is eectively locally nite. PPP Suppose that E > 0. Then there is ameasurable compact set K E such that K > 0. As in the argument for (a) above, there is an open setG of nite measure including K, so that (E G) > 0. QQQ(g) Corresponding to (a) above, we have the following fact. If is a measure on a topological space andf L0() is locally integrable, then _K fd is nite for every compact K X, because K can be coveredby a nite family of open sets G such that_G[f[d < .(h) If is a locally nite measure on a topological space X, and f Lp() for some p [1, ], then f islocally integrable; this is because_G[f[ _E f |f|p|E|q is nite whenever G E and E < , where1p + 1q = 1, by Holders inequality (244Eb).(i) If (X, T) is a completely regular space and is a locally nite topological measure on X, then theset of open sets with negligible boundaries is a base for T. PPP If x G T, let H G be an open set ofnite measure containing x, and f : X [0, 1] a continuous function such that f(x) = 1 and f(y) = 0 fory X H. Then f1[] : 0 < < 1 is an uncountable disjoint family of measurable subsets of H, sothere must be some ]0, 1[ such that f1[] is negligible. Set U = y : f(y) > ; then U is an openneighbourhood of x included in G and U f1[] is negligible. QQQ411H Two particularly important combinations of the properties above are the following.Denitions (a) A quasi-Radon measure space is a topological measure space (X, T, , ) such that (i)(X, , ) is complete and locally determined (ii) is -additive, inner regular with respect to the closed setsand eectively locally nite.(b) A Radon measure space is a topological measure space (X, T, , ) such that (i) (X, , ) iscomplete and locally determined (ii) T is Hausdor (iii) is locally nite and inner regular with respect tothe compact sets.411I Remarks(a) You may like to seek your own proof that a Radon measure space is always quasi-Radon, before looking it up in 416 below.(b) Note that a measure on Euclidean space Rris a Radon measure on the denition above i it is aRadon measure as described in 256Ad. PPP In 256Ad, I said that a measure on Rris Radon if it is alocally nite complete topological measure, inner regular with respect to the compact sets. (The denitionof locally nite in 256A was not the same as the one above, but I have already covered this point in 411Gb.)So the only thing to add is that is necessarily locally determined, because it is -nite (256Ba). QQQ411J The following special types of inner regularity are of sucient importance to have earned separatenames.Denitions (a) If (X, T) is a topological space, I will say that a measure on X is tight if it is innerregular with respect to the closed compact sets.(b) If (X, T, , ) is a topological measure space, I will say that is completion regular if it is innerregular with respect to the zero sets (denition: 3A3Pa).411K Borel and Baire measures If (X, T) is a topological space, I will call a measure with domain(exactly) the Borel -algebra of X (4A3A) a Borel measure on X, and a measure with domain (exactly)the Baire -algebra of X (4A3K) a Baire measure on X.Of course a Borel measure is a topological measure in the sense of 411A. On a metric space, the Boreland Baire measures coincide (4A3Kb). The most important measures in this chapter will be c.l.d. versionsof Borel measures.16 Topologies and measures I 411L411L When we come to look at functions dened on a topological measure space, we shall have to relateideas of continuity and measurability. Two basic concepts are the following.Denition Let X be a set, a -algebra of subsets of X and (Y, S) a topological space. I will say that afunction f : X Y is measurable if f1[G] for every open set G Y .Remarks (a) Note that a function f : X R is measurable on this denition (when R is given its usualtopology) i it is measurable according to the familiar denition in 121C, which asks only that sets of theform x : f(x) < should be measurable (121Ef).(b) For any topological space (Y, S), a function f : X Y is measurable i f is (, B(Y ))-measurable,where B(Y ) is the Borel -algebra of Y (4A3Cb).411M Denition Let (X, , ) be a measure space, T a topology on X, and (Y, S) another topologicalspace. I will say that a function f : X Y is almost continuous or Lusin measurable if is innerregular with respect to the family of subsets A of X such that fA is continuous.411N Finally, I introduce some terminology to describe ways in which (sometimes) measures can belocated in one part of a topological space rather than another.Denitions Let (X, , ) be a measure space and T a topology on X.(a) I will call a set A X self-supporting if (A G) > 0 for every open set G such that A G isnon-empty. (Such sets are sometimes called of positive measure everywhere.)(b) A support of is a closed self-supporting set F such that X F is negligible.(c) Note that can have at most one support. PPP If F1, F2 are supports then (F1F2) (XF2) = 0so F1 F2 must be empty. Similarly, F2 F1 = , so F1 = F2. QQQ(d) If is a -additive topological measure it has a support. PPP Let ( be the family of negligible opensets, and F the closed set X

(. Then ( is an upwards-directed family in T and

( T , so(X F) = (

() = supG G = 0.If G is open and G F ,= then G / ( so (G F) = (G F) = G > 0; thus F is self-supporting andis the support of . QQQ(e) Let X and Y be topological spaces with topological measures , respectively and a continuousinverse-measure-preserving function f : X Y . Suppose that has a support E. Then f[E] is the supportof . PPP We have only to observe that for an open set H YH > 0 f1[H] > 0 f1[H] E ,= H f[E] ,= H f[E] ,= . QQQ(f ) is strictly positive (with respect to T) if G > 0 for every non-empty open set G X, that is,X itself is the support of .*(g) If (X, T) is a topological space, and is a strictly positive -nite measure on X such that the domain of includes a -base | for T, then X is ccc. PPPLet En)nN be a sequence of sets of nite measure coveringX. Let ( be a disjoint family of non-empty open sets. For each G G, take UG | such that UG G;then UG > 0, so there is an n(G) such that (En(G) UG) > 0. Now GG,n(G)=k (Ek UG) Ek isnite for every k, so G : n(G) = k must be countable and ( is countable. QQQ411O Example Lebesgue measure on Rris a Radon measure (256Ha); in particular, it is locally niteand tight. It is therefore -additive and eectively locally nite (411E, 411Gf). It is completion regular(because every compact set is a zero set, see 4A2Lc), outer regular with respect to the open sets (134F) andstrictly positive.411Q Denitions 17411P Example: Stone spaces (a) Let (Z, T, , ) be the Stone space of a semi-nite measure algebra(A, ), so that (Z, T) is a zero-dimensional compact Hausdor space, (Z, , ) is complete and semi-nite,the open-and-closed sets are measurable, the negligible sets are the nowhere dense sets, and every measurableset diers by a nowhere dense set from an open-and-closed set (311I, 321K, 322Bd, 322Qa).(b) is inner regular with respect to the open-and-closed sets (322Qa); in particular, it is completionregular and tight. Consequently it is -additive (411E).(c) is strictly positive, because the open-and-closed sets form a base for T (311I) and a non-empty open-and-closed set has non-zero measure. is eectively locally nite. PPP Suppose that E is not negligible.There is a measurable set F E such that 0 < F < ; now there is a non-empty open-and-closed set Gincluded in F, in which case G < and (E G) > 0. QQQ(d) The following are equiveridical, that is, if one is true so are the others:(i) (A, ) is localizable;(ii) is strictly localizable;(iii) is locally determined;(iv) is a quasi-Radon measure.PPP The equivalence of (i)-(iii) is Theorem 322N. (iv)(iii) is trivial. If one, therefore all, of (i)-(iii) are true,then is a topological measure, because if G Z is open, then G is open-and-closed, by 314S, thereforemeasurable, and G G is nowhere dense, therefore also measurable. We know already that is complete,eectively locally nite and -additive, so that if it is also locally determined it is a quasi-Radon measure.QQQ(e) The following are equiveridical:(i) is a Radon measure;(ii) is totally nite;(iii) is locally nite;(iv) is outer regular with respect to the open sets.PPP (ii)(iv) If is totally nite and E , then for any > 0 there is a closed set F Z E such thatF (Z E) , and now G = Z F is an open set including E with G E + . (iv)(iii) Supposethat is outer regular with respect to the open sets, and z Z. Because Z is Hausdor, z is closed. If itis open it is measurable, and because is semi-nite it must have nite measure. Otherwise it is nowheredense, therefore negligible, and must be included in open sets of arbitrarily small measure. Thus in bothcases z belongs to an open set of nite measure; as z is arbitrary, is locally nite. (iii)(ii) Becasue Zis compact, this is a consequence of 411Ga. (i)(iii) is part of the denition of Radon measure. Finally,(ii)+(iii)(i), again directly from the denition and the facts set out in (a)-(b) above. QQQ411Q Example: Dieudonnes measure Recall that a set E 1 is a Borel set i either E or itscomplement includes a conal closed set (4A3J). So we may dene a Borel measure on 1 by saying thatE = 1 if E includes a conal closed set and E = 0 if E is disjoint from a conal closed set. If E is disjointfrom some conal closed set, so is any subset of E, so is complete. Since takes only the values 0 and 1,it is a purely atomic probability measure. is a topological measure; being totally nite, it is surely locally nite and eectively locally nite. Itis inner regular with respect to the closed sets (because if E > 0, there is a conal closed set F E,and now F is a closed set with F = E), therefore outer regular with respect to the open sets. It is not-additive (because = [0, [ is an open set of zero measure for every < 1, and the union of these sets isa measurable open set of measure 1). is not completion regular, because the set of countable limit ordinals is a closed set (4A1Bb) whichdoes not include any uncountable zero set (see 411Ra below).The only self-supporting subset of 1 is the empty set (because there is a cover of 1 by negligible opensets). In particular, does not have a support.Remark There is a measure of this type on any ordinal of uncountable conality; see 411Xj.18 Topologies and measures I 411R411R Example: The Baire -algebra of 1 The Baire -algebra Ba(1) of 1 is the countable-cocountable algebra (4A3P). The countable-cocountable measure on 1 is therefore a Baire measure onthe denition of 411K. Since all sets of the form ], 1[ are zero sets, is inner regular with respect to thezero sets and outer regular with respect to the cozero sets. Since sets of the form [0, [ (= ) form a coverof 1 by measurable open sets of zero measure, is not -additive.411X Basic exercises (a) Let (X, , ) be a totally nite measure space and T a topology on X. Showthat is inner regular with respect to the closed sets i it is outer regular with respect to the open sets,and is inner regular with respect to the zero sets i it is outer regular with respect to the cozero sets.(b) Let be a Radon measure on Rr, where r 1, and f L0(). Show that f is locally integrable inthe sense of 411Fc i it is locally integrable in the sense of 256E, that is,_E fd < for every bounded setE Rr.(c) Let be a measure on a topological space, its completion and its c.l.d. version. Show that islocally nite i is locally nite, and in this case is locally nite.>>>(d) Let be an eectively locally nite measure on a topological space X. (i) Show that the completionand c.l.d. version of are eectively locally nite. (ii) Show that if is complete and locally determined, thenthe union of the measurable open sets of nite measure is conegligible. (iii) Show that if X is hereditarilyLindelof then must be -nite.(e) Let X be a topological space and a measure on X. Let U L0() be the set of equivalence classesof locally integrable functions in L0(). Show that U is a solid linear subspace of L0(). Show that if islocally nite then U includes Lp() for every p [0, ].(f ) Let X be a topological space. (i) Let , be two totally nite Borel measures which agree on theclosed sets. Show that they are equal. (Hint: 136C.) (ii) Let , be two totally nite Baire measures whichagree on the zero sets. Show that they are equal.(g) Let (X, T) be a topological space, a measure on X, and Y a subset of X; let TY , Y be the subspacetopology and measure. Show that if is a topological measure, or locally nite, or a Borel measure, so isY .(h) Let (Xi, i, i))iI be a family of measure spaces, with direct sum (X, , ); suppose that we aregiven a topology Ti on each Xi, and let T be the disjoint union topology on X. Show that is a topologicalmeasure, or locally nite, or eectively locally nite, or a Borel measure, or a Baire measure, or strictlypositive, i every i is.(i) Let (X, , ) and (Y, T, ) be two measure spaces, with c.l.d. product measure on X Y . Supposewe are given topologies T, S on X, Y respectively, and give X Y the product topology. Show that islocally nite, or eectively locally nite, if and are.(j) Let be any cardinal of uncountable conality (denition: 3A1Fb). Show that there is a completetopological probability measure on dened by saying that E = 1 if E includes a conal closed set in ,0 if E is disjoint from some conal closed set. Show that is inner regular with respect to the closed setsbut is not completion regular.411Y Further exercises (a) Show that a function f : RrRsis measurable i it is almost continuous(where Rris endowed with Lebesgue measure and its usual topology, of course). (Hint: 256F.)(b) Let (X, ) be a metric space, r 0, and write Hr for r-dimensional Hausdor measure on X (264K,471). (i) Show that Hr is a topological measure, outer regular with respect to the Borel sets. (ii) Showthat if X is complete then the c.l.d. version of Hr is tight, therefore completion regular.412A Inner regularity 19(c) Let (X, T, , ) be a topological measure space. Set c = E : E X, (E) = 0, where E isthe boundary of A. (i) Show that c is a subalgebra of TX, and that every member of c is measured bythe completion of . (c is sometimes called the Jordan algebra of (X, T, , ). Do not confuse with theJordan algebras of abstract algebra.) (ii) Suppose that is complete and totally nite and inner regularwith respect to the closed sets, and that T is normal. Show that E : E c is dense in the measurealgebra of endowed with its usual topology. (Hint: if f : X R is continuous, then x : f(x) cfor all but countably many .) (iii) Suppose that is a quasi-Radon measure and T is completely regular.Show that E : E c is dense in the measure algebra of . (Hint: 414Aa.)411 Notes and comments Of course the list above can give only a rough idea of the ways in whichtopologies and measures can interact. In particular I have rather arbitrarily given a sort of priority to threeparticular relationships between the domain of a measure and the topology: topological measure space(in which includes the Borel -algebra), Borel measure (in which is precisely the Borel -algebra) andBaire measure (in which is the Baire -algebra).Abstract topological measure theory is a relatively new subject, and there are many technical questionson which dierent authors take dierent views. For instance, the phrase Radon measure is commonly usedto mean what I would call a tight locally nite Borel measure (cf. 416F); and some writers enlarge thedenition of topological measure to include Baire measures as dened above.I give very few examples at this stage, two drawn from the constructions of Volumes 1-3 (Lebesguemeasure and Stone spaces, 411O-411P) and one new one (Dieudonnes measure, 411Q), with a glance atthe countable-cocountable measure of 1 (411R). The most glaring omission is that of the product measureson 0, 1Iand [0, 1]I. I pass these by at the moment because a proper study of them requires rather morepreparation than can be slipped into a parenthesis. (I return to them in 416U.) I have also omitted anydiscussion of measurable and almost continuous functions, except for a reference to a theorem in Volume 2(411Ya), which will have to be repeated and amplied later on (418). There is an obvious complementaritybetween the notions of inner and outer regularity (411B, 411D), but it works well only for totally nitespaces (411Xa); in other cases it may not be obvious what will happen (411O, 411Pe, 412W).412 Inner regularityAs will become apparent as the chapter progresses, the concepts introduced in 411 are synergic; theirmost interesting manifestations are in combinations of various kinds. Any linear account of their propertieswill be more than usually like a space-lling curve. But I have to start somewhere, and enough results canbe expressed in terms of inner regularity, more or less by itself, to be a useful beginning.After a handful of elementary basic facts (412A) and a list of standard applications (412B), I give someuseful sucient conditions for inner regularity of topological and Baire measures (412D, 412E, 412G), basedon an important general construction (412C). The rest of the section amounts to a review of ideas from Vol-ume 2 and Chapter 32 in the light of the new concept here. I touch on completions (412H), c.l.d. versions andcomplete locally determined spaces (412H, 412J, 412L), strictly localizable spaces (412I), inverse-measure-preserving functions (412K, 412M), measure algebras (412N), subspaces (412O, 412P), indenite-integralmeasures (412Q) and product measures (412R-412V), with a brief mention of outer regularity (412W); mostof the hard work has already been done in Chapters 21 and 25.412A I begin by repeating a lemma from Chapter 34, with some further straightforward facts.Lemma (a) Let (X, , ) be a measure space and / a family of sets such thatwhenever E and E > 0 there is a K / such that K E and K > 0.Then whenever E there is a countable disjoint family Ki)iI in / such that Ki E for every iand

iI Ki = E. If moreover() K Kt / whenever K, Kt are disjoint members of /,then is inner regular with respect to /. If

iI Ki / for every countable disjoint family Ki)iI in /,then for every E there is a K / such that K E and K = E.20 Topologies and measures 412A(b) Let (X, , ) be a measure space, T a -subalgebra of , and / a family of sets. If is inner regularwith respect to T and T is inner regular with respect to /, then is inner regular with respect to /.(c) Let (X, , ) be a semi-nite measure space and /n)nN a sequence of families of sets such that isinner regular with respect to /n and() if Ki)iN is a non-increasing sequence in /n, then

iNKi /nfor every n N. Then is inner regular with respect to

nN/n.proof (a) This is 342B-342C.(b) If E and < E, there are an F T such that F E and F > , and a K /T such thatK F and K .(c) Suppose that E and that 0 < E. Because is semi-nite, there is an F such thatF E and < F < (213A). Choose Ki)iN inductively, as follows. Start with K0 = F. Given thatKi and < Ki, then let ni N be such that 2ni(i + 1) is an odd integer, and choose Ki+1 /nisuch that Ki+1 Ki and Ki+1 > ; this will be possible because is inner regular with respect to /ni.Consider K =

iNKi. Then K E and K = limiKi . But alsoK =

jNK2n(2j+1) /nbecause K2n(2j+1))jN is a non-increasing sequence in /n, for each n. So K

nN/n. As E and arearbitrary, is inner regular with respect to

nN/n.412B Corollary Let (X, , ) be a measure space and T a topology on X. Suppose that / iseither the family of Borel subsets of Xor the family of closed subsets of Xor the family of compact subsets of Xor the family of zero sets in X,and suppose that whenever E and E > 0 there is a K / such that K E and K > 0. Then is inner regular with respect to /.proof In every case, / satises the condition () of 412Aa.412C The next lemma provides a particularly useful method of proving that measures are inner regularwith respect to well-behaved families of sets.Lemma Let (X, , ) be a semi-nite measure space, and suppose that / and / are such that / ,() K Kt / whenever K, Kt /,()

nNKn / for every sequence Kn)nN in /,X A / for every A /,whenever A /, F and (A F) > 0, there is a K / / such that K A and(K F) > 0.Let T be the -subalgebra of generated by /. Then T is inner regular with respect to /.proof (a) Write A for the measure algebra of (X, , ), and L = /T, so that L is also closed under niteunions and countable intersections. SetH = E : E , supL1,LE L = E in A,Tt = E : E H, X E H,so that the last two conditions tell us that / Tt.(b) The intersection of any sequence in H belongs to H. PPP Let Hn)nN be a sequence in H withintersection H. Write An for L : L L, L Hn A for each n N. Since A is weakly (, )-distributive (322F), An is upwards-directed, and sup An = Hn for each n N,412F Inner regularity 21H = infnNHn(because F F : A is sequentially order-continuous, by 321H)= infnNsup An = sup infnNan : an An for every n N(316J)= sup(

nNLn) : Ln L, Ln Hn for every n N L : L L, L H(by ()) H,and H H. QQQ(c) The union of any sequence in H belongs to H. PPP If Hn)nN is a sequence in H with union H thensupL1,LH L supnN supL1,LEn L = supnNHn = H,so H H. QQQ(d) Tt is a -subalgebra of . PPP (i) and X belong to / H, so Tt. (ii) Obviously X E Ttwhenever E Tt. (iii) If En)nN is a sequence in Tt with union E then E H, by (c); but alsoX E =

nN(X En) belongs to H, by (b). So E Tt. QQQ(e) Accordingly T Tt, and E = supL1,LE L for every E T. It follows at once that if E T andE > 0, there must be an L L such that L E and L > 0; since () is true, and L T, we can apply412Aa to see that T is inner regular with respect to L, therefore with respect to /.412D As corollaries of the last lemma I give two-and-a-half basic theorems.Theorem Let (X, T) be a topological space and a semi-nite Baire measure on X. Then is inner regularwith respect to the zero sets.proof Write for the Baire -algebra of X, the domain of , / for the family of zero sets, and / for/ X K : K /. Since the union of two zero sets is a zero set (4A2C(b-ii)), the intersection of asequence of zero sets is a zero set (4A2C(b-iii)), and the complement of a zero set is the union of a sequenceof zero sets (4A2C(b-vi)), the conditions of 412C are satised; and as the -algebra generated by / is just, is inner regular with respect to /.412E Theorem Let (X, T) be a perfectly normal topological space (e.g., any metrizable space). Thenany semi-nite Borel measure on X is inner regular with respect to the closed sets.proof Because the Baire and Borel -algebras are the same (4A3Kb), this is a special case of 412D.412F Lemma Let (X, , ) be a measure space and T a topology on X such that is eectively locallynite with respect to T. ThenE = sup(E G) : G is a measurable open set of nite measurefor every E .proof Apply 412Aa with / the family of subsets of measurable open sets of nite measure.22 Topologies and measures 412G412G Theorem Let (X, , ) be a measure space with a topology T such that is eectively locallynite with respect to T and is the -algebra generated by T . IfG = supF : F is closed, F Gfor every measurable open set G of nite measure, then is inner regular with respect to the closed sets.proof In 412C, take / to be the family of measurable closed subsets of X, and / to be the family ofmeasurable sets which are either open or closed. If G T, F and (G F) > 0, then there is anopen set H of nite measure such that (H G F) > 0, because is eectively locally nite; now thereis a K / such that K H G and K > (H G) (H G F), so that (K F) > 0. This is theonly non-trivial item in the list of hypotheses in 412C, so we can conclude that T is inner regular withrespect to /, where T is the -algebra generated by /; but of course this is just .Remark There is a similar result in 416F(iii) below.412H Proposition Let (X, , ) be a measure space and / a family of sets.(a) If is inner regular with respect to /, so are its completion (212C) and c.l.d. version (213E).(b) Now suppose that()

nNKn / whenever Kn)nN is a non-increasing sequence in /.If either is inner regular with respect to / or is semi-nite and is inner regular with respect to /,then is inner regular with respect to /.proof (a) If F belongs to the domain of , then there is an E such that E F and (F E) = 0. Soif 0 < F = E, there is a K / such that K E F and K = K .If H belongs to the domain of and 0 < H, there is an E such that E < and (EH) > (213D). Now there is a K / such that K E H and K . As K < , K = K .(b) Write for whichever of , is supposed to be inner regular with respect to /. Then is innerregular with respect to (212Ca, 213Fc), so is inner regular with respect to / (412Ac). Also extends (212D, 213Hc). Take E and < E = E. Then there is a K / such that K E and < K = K. As E and are arbitrary, is inner regular with respect to /.412I Lemma Let (X, , ) be a strictly localizable measure space and / a family of sets such thatwhenever E and E > 0 there is a K / such that K E and K > 0.(a) There is a decomposition Xi)iI of X such that at most one Xi does not belong to /, and thatexceptional one, if any, is negligible.(b) There is a disjoint family L / such that A =

L1(A L) for every A X.(c) If is -nite then the family Xi)iI of (a) and the set L of (b) can be taken to be countable.proof (a) Let Ej)jJ be any decomposition of X. For each j J, let /j be a maximal disjoint subset ofK : K / , K Ej, K > 0.Because Ej < , /j must be countable. Set Etj = Ej

/j. By the maximality of /j, Etj cannot includeany non-negligible set in / ; but this means that Etj = 0. Set Xt =

jJ Etj. ThenXt =

jJ (Xt Ej) =

jJ Etj = 0.Note that if j, jt J are distinct, and K /j, Kt /j , then KKt = ; thus L =

jJ /j is disjoint.Let Xi)iI be any indexing of Xt L. This is a partition (that is, disjoint cover) of X into sets of nitemeasure. If E X and E Xi for every i I, then for every j JE Ej = (E Xt Ej)

KKjE Kbelongs to , so that E andE =

jJ

KKj(E K) =

iI (E Xi).Thus Xi)iI is a decomposition of X, and it is of the right type because every Xi but one belongs to L /.412K Inner regularity 23(b) If now A X is any set,A = AA =

iI A(A Xi) =

iI (A Xi)by 214Ia, writing A for the subspace measure on A. So we haveA = (A Xt) +

L1(A L) =

L1(A L),while L / is disjoint.(c) If is -nite we can take J to be countable, so that I and L will also be countable.412J Proposition Let (X, , ) be a complete locally determined measure space, and / a family of setssuch that is inner regular with respect to /.(a) If E X is such that E K for every K / , then E .(b) If E X is such that E K is negligible for every K / , then E is negligible.(c) For any A X, A = supKK(A K).(d) Let f be a non-negative [0, ]-valued function dened on a subset of X. If_K f is dened in [0, ]for every K /, then_ f is dened and equal to supKK_K f.(e) If f is a -integrable function and > 0, there is a K / such that_X\K[f[ .Remark In (c), we must interpret sup as 0 if / = .proof (a) If F and F < , then E F . PPP If F = 0, this is trivial, because is complete andE F is negligible. Otherwise, there is a sequence Kn)nN in / such that Kn F for each n andsupnNKn = F. Now E F

nNKn is negligible, therefore measurable, while E Kn is measurablefor every n N, by hypothesis; so E F is measurable. QQQ As is locally determined, E , as claimed.(b) By (a), E ; and because is inner regular with respect to /, E must be 0.(c) Let A be the subspace measure on A. Because is complete and locally determined, A is semi-nite(214Ic). So if 0 < A = AA, there is an H A such that AH is dened, nite and greater than .Let E be a measurable envelope of H (132Ee), so that E = H > . Then there is a K / such that K E and K . In this case(A K) (H K) = (E K) = K .As is arbitrary,A supKK(A K);but the reverse inequality is trivial, so we have the result.(d) Applying (b) with E = X domf, we see that f is dened almost everywhere in X. Applying (a)with E = x : x domf, f(x) for each R, we see that f is measurable. So_ f is dened in [0, ],and of course_ f supKK_K f. If ; taking E = x : g(x) > 0, there is a K / such that K E and (E K)|g| _ g,so that_K f _K g . As is arbitrary,_ f = supKK_K f.(e) By (d), there is a K / such that_K[f[ _ [f[ .Remark See also 413F below.412K Proposition Let (X, , ) be a complete locally determined measure space, (Y, T, ) a measurespace and f : X Y a function. Suppose that / T is such that(i) is inner regular with respect to /;(ii) f1[K] and f1[K] = K for every K /;(iii) whenever E and E > 0 there is a K / such that K < and (E f1[K]) > 0.Then f is inverse-measure-preserving for and .proof (a) If F T, E and E < , then Ef1[F] . PPP Let H1, H2 be measurable envelopesfor E f1[F] and E f1[F] respectively. ??? If (H1 H2) > 0, there is a K / such that K is nite24 Topologies and measures 412Kand (H1 H2 f1[K]) > 0. Because is inner regular with respect to /, there are K1, K2 / suchthat K1 K F, K2 K F andK1 +K2 > (K F) +(K F) (H1 H2 f1[K]) = K (H1 H2 f1[K]).Now(H1 f1[K2]) = (E f1[F] f1[K2]) = 0,(H2 f1[K1]) = (E f1[K1] f1[F]) = 0,so (H1 H2 f1[K1 K2]) = 0 and(H1 H2 f1[K]) (f1[K] f1[K1 K2])= f1[K] f1[K1] f1[K2]= K K1 K2 < (H1 H2 f1[K]),which is absurd. XXXNow (E H1) (E f1[F]) H1H2 is negligible, therefore measurable (because is complete), andE f1[F] , as claimed. QQQ(b) It follows (because is locally determined) that f1[F] for every F T.(c) If F T and F = 0 then f1[F] = 0. PPP??? Otherwise, there is a K / such that K < and0 < (f1[F] f1[K]) = f1[F K].Let Kt / be such that Kt K F and Kt > K f1[F K]. Then f1[Kt] f1[F K] = , soK = f1[K] f1[Kt] +f1[F K] > Kt +K Kt = K,which is absurd. XXXQQQ(d) Finally, f1[F] = F for every F T. PPP Let Ki)iI be a countable disjoint family in / such thatKi F for every i and

iI Ki = F (412Aa). Set Ft = F

iI Ki. Thenf1[F] = f1[Ft] +

iI f1[Ki] = f1[Ft] +

iI Ki = f1[Ft] +F.If F = then surely f1[F] = = F. Otherwise, Ft = 0 so f1[Ft] = 0 (by (c)) and againf1[F] = F. QQQThus f is inverse-measure-preserving.412L Corollary Let X be a set and / a family of subsets of X. Suppose that , are two completelocally determined measures on X, with domains including /, agreeing on /, and both inner regular withrespect to /. Then they are identical (and, in particular, have the same domain).proof Apply 412K with X = Y and f the identity function to see that extends ; similarly, extends and the two measures are the same.412M Corollary Let (X, , ) be a complete probability space, (Y, T, ) a probability space and f :X Y a function. Suppose that whenever F T and F > 0 there is a K T such that K F, K > 0,f1[K] and f1[K] K. Then f is inverse-measure-preserving.proof Set / = K : K T, f1[K] , f1[K] K. Then / is closed under countable disjointunions and includes /, so for every F T there is a K / such that K F and K = F, by 412Aa.But this means that f1[K] = K for every K /. PPP There is a Kt / such that Kt Y K andKt = 1 K; but in this casef1[Kt] +f1[K] 1 = Kt +K,so f1[K] must be equal to K. QQQ Moreover, there is a K / such that K = Y = 1, sof1[K] = X = 1 and (E f1[K]) > 0 whenever E > 0. Applying 412K to / we have the result.412Q Inner regularity 25412N Lemma Let (X, , ) be a measure space and / a family of subsets of X such that is innerregular with respect to /. ThenE = supK : K / , K Ein the measure algebra A of , for every E . In particular, K : K / is order-dense in A; and if/ is closed under nite unions, then K : K / is topologically dense in A for the measure-algebratopology.proof ??? If E ,= supK : K / , K E, there is a non-zero a A such that a E \ K wheneverK / and K E. Express a as F where F E. Then F > 0, so there is a K / such thatK F and K > 0. But in this case 0 ,= K a, while K E. XXXIt follows at once that D = K : K / is order-dense. If / is closed under nite unions, anda A, then Da = d : d D, d a is upwards-directed and has supremum a, so a Da D (323D(a-ii)).412O Lemma Let (X, , ) be a measure space and / a family of subsets of X such that is innerregular with respect to /.(a) If E , then the subspace measure E (131B) is inner regular with respect to /.(b) Let Y X be any set such that the subspace measure Y (214A-214B) is semi-nite. Then Y isinner regular with respect to /Y = K Y : K /.proof (a) This is elementary.(b) Suppose that F belongs to the domain Y of Y and 0 < YF. Because Y is semi-nite thereis an Ft Y such that Ft F and < YFt < . Let E be a measurable envelope for Ft with respectto , so thatE = Ft = YFt > .There is a K / such that K E and K , in which case K Y /Y Y andY (K Y ) = (K Y ) = (K Ft) = (K E) = K .As F and are arbitrary, Y is inner regular with respect to /Y .Remark Recall from 214I that if (X, , ) has locally determined negligible sets (in particular, is eitherstrictly localizable or complete and locally determined), then all its subspaces are semi-nite.412P Proposition Let (X, , ) be a measure space, T a topology on X and Y a subset of X; write TYfor the subspace topology of Y and Y for the subspace measure on Y . Suppose that either Y or Yis semi-nite.(a) If is a topological measure, so is Y .(b) If is inner regular with respect to the Borel sets, so is Y .(c) If is inner regular with respect to the closed sets, so is Y .(d) If is inner regular with respect to the zero sets, so is Y .(e) If is eectively locally nite, so is Y .proof (a) is an immediate consequence of the denitions of subspace measure, subspace topology andtopological measure. The other parts follow directly from 412O if we recall that(i) a subset of Y is Borel for TY whenever it is expressible as Y E for some Borel set E X (4A3Ca);(ii) a subset of Y is closed in Y whenever it is expressible as Y F for some closed set F X;(iii) a subset of Y is a zero set in Y whenever it is expressible as Y F for some zero set F X(4A2C(b-v));(iv) is eectively locally nite i it is inner regular with respect to subsets of open sets of nite measure.412Q Proposition Let (X, , ) be a measure space, and an indenite-integral measure over (de-nition: 234B). If is inner regular with respect to a family / of sets, so is .proof Because and its completion give the same integrals, is an indenite-integral measure over ;and as is still inner regular with respect to / (412H), we may suppose that itself is complete. Let f26 Topologies and measures 412Qbe a Radon-Nikod ym derivative of with respect to ; by 234Ca, we may suppose that f : X [0, [ is-measurable.Suppose that F dom() and that < F. Set G = x : f(x) > 0, so that F G (234D). Forn N, set Hn = x : x F, 2n f(x) 2n, so that Hn andF =_ f Fd = limn_ f Hnd.Let n N be such that_ f Hnd > .If Hn = , there is a K / such that K Hn and K 2n, so that K . If Hn is nite, thereis a K / such that 2n(HnK) _ f Hnd , so that_ f (Hn K)d + _ f Hn andK =_ f K d . Thus in either case we have a K / such that K F and K ; as F and are arbitrary, is inner regular with respect to /.412R Lemma Let (X, , ) and (Y, T, ) be measure spaces, with c.l.d. product space (X Y, , )(251F). Suppose that / TX, L TY , / T(X Y ) are such that(i) is inner regular with respect to /;(ii) is inner regular with respect to L;(iii) K L / for all K /, L L;(iv) M Mt / whenever M, Mt /;(v)

nNMn / for every sequence Mn)nN in /.Then is inner regular with respect to /.proof Write / = E Y : E X F : F T. Then if V /, W and (W V ) > 0, there isan M / / such that M W and (M V ) > 0. PPP Suppose that V = E Y where E . Theremust be E0 and F0 T, both of nite measure, such that (W V (E0F0)) > 0 (251F). Now thereis a K / such that K E E0 and ((E E0) K) F0 < (W V (E0F0)); but this means thatM = KY is included in V and (W M) > 0. Reversing the roles of the coordinates, the same argumentdeals with the case in which V = X F for some F T. QQQBy 412C, 0 = T is inner regular with respect to /. But is inner regular with respect to T(251Ib) so is also inner regular with respect to / (412Ab).412S Proposition Let (X, , ) and (Y, T, ) be measure spaces, with c.l.d. product space (XY, , ).Let T, S be topologies on X and Y respectively, and give X Y the product topology.(a) If and are inner regular with respect to the closed sets, so is .(b) If and are tight (that is, inner regular with respect to the closed compact sets), so is .(c) If and are inner regular with respect to the zero sets, so is .(d) If and are inner regular with respect to the Borel sets, so is .(e) If and are eectively locally nite, so is .proof We have only to read the conditions (i)-(v) of 412R carefully and check that they apply in eachcase. (In part (e), recall that eectively locally nite is the same thing as inner regular with respect tothe subsets of open sets of nite measure.)412T Lemma Let (Xi, i, i))iI be a family of probability spaces, with product probability space(X, , ) (254). Suppose that /i TXi, / TX are such that(i) i is inner regular with respect to /i for each i I;(ii) 1i [K] / for every i I, K /i, writing i(x) = x(i) for x X;(iii) M Mt / whenever M, Mt /;(iv)

nNMn / for every sequence Mn)nN in /.Then is inner regular with respect to /.proof The argument is nearly identical to that of 412R. Write / = 1i [E] : i I, E i. Then ifV /, W and (W V ) > 0, express V as 1i [E], where i I and E i, and take K /i suchthat K E and i(E K) < (W V ); then M = 1i [K] belongs to //, is included in W, and meetsV in a non-negligible set. So, just as in 412R, the conditions of 412C are met.It follows that 0 =

iIi is inner regular with respect to /. But is the completion of 0 (254Fd,254Ff), so is also inner regular with respect to / (412Ha).*412W Inner regularity 27412U Proposition Let (Xi, i, i))iI be a family of probability spaces, with product probability space(X, , ). Suppose that we are given a topology Ti on each Xi, and let T be the product topology on X.(a) If every i is inner regular with respect to the closed sets, so is .(b) If every i is inner regular with respect to the zero sets, so is .(c) If every i is inner regular with respect to the Borel sets, so is .proof This follows from 412T just as 412S follows from 412R.412V Corollary Let (Xi, i, i))iI be a family of probability spaces, with product probability space(X, , ). Suppose that we are given a Hausdor topology Ti on each Xi, and let T be the product topologyon X. Suppose that every i is tight, and that Xi is compact for all but countably many i I. Then istight.proof By 412Ua, is inner regular with respect to the closed sets. If W and < W, let V Wbe a measurable closed set such that V > . Let J be the set of those i I such that Xi is notcompact; we are supposing that J is countable. Let i)iJ be a family of strictly positive real numberssuch that iJ j V (4A1P). For each i J, let Ki Xi be a compact measurable set such thati(Xi Ki) i; and for i I J, set Ki = Xi. Then K =

iI Ki is a compact measurable subset of X,and(X K)

iJ (Xi Ki) V ,so (K V ) ; while K V is a compact measurable subset of W. As W and are arbitrary, is tight.*412W Outer regularity I have already mentioned the complementary notion of outer regularity(411D). In this book it will not be given much prominence. It is however a useful tool when dealing withLebesgue measure (see, for instance, the proof of 225K), for reasons which the next proposition will makeclear.Proposition Let (X, , ) be a measure space and T a topology on X.(a) Suppose that is outer regular with respect to the open sets. Then for any integrable functionf : X [0, ] and > 0, there is a lower semi-continuous measurable function g : X [0, ] such thatf g and_ g +_ f.(b) Now suppose that there is a sequence of measurable open sets of nite measure covering X. Thenthe following are equiveridical:(i) is inner regular with respect to the closed sets;(ii) is outer regular with respect to the open sets;(iii) for any measurable set E X and > 0, there are a measurable closed set F E and a measurableopen set H E such that (H F) ;(iv) for every measurable function f : X [0, [ and > 0, there is a lower semi-continuous measurablefunction g : X [0, ] such that f g and_ g f ;(v) for every measurable function f : X R and > 0, there is a lower semi-continuous measurablefunction g : X ], ] such that f g and x : g(x) f(x) + .proof (a) Let ]0, 1] be such that (7 +_ fd) . For n Z, set En = x : (1 + )n f(x) 0. For each n N, let Hn Gn En be an open set such that (Hn En) 2n2;then H =

nNHn is a measurable open set including E, and (H E) 12. Now repeat the argumenton X E to nd a measurable closed set F E such that (E F) 12.(iii)(iv) Assume (iii), and let f : X [0, [ be a measurable function, > 0. Set n = 2n/(16 +4Gn) for each n N. For k N set Ek =

nNx : x Gn, kn f(x) < (k +1)n, and choose an openset Hk Ek such that (Hk Ek) 2k. Setg = supk,nN(k + 1)n(Gn