http://dx.doi.org/10.1090/gsm/076 Measur e Theor y an d Integratio n This page intentionally left blank Measur e Theor y an d Integratio n Michael E.Taylor Graduate Studies in Mathematics Volume 76 M^^t| American Mathematical Society ^MMOT Providence, Rhode Island Editorial Board David Cox Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair) 2000 Mathematics Subject Classification. Primary 28-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-76 Library of Congress Cataloging-in-Publication Data Taylor, Michael Eugene, 1946- Measure theory and integration / Michael E. Taylor. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 76) Includes bibliographical references. ISBN-13: 978-0-8218-4180-8 1. Measure theory. 2. Riemann integrals. 3. Convergence. 4. Probabilities. I. Title. II. Series. QA312.T387 2006 515/.42—dc22 2006045635 Copying and reprinting. 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The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 98765432 11 Contents Introduction Chapter 1. The Riemann Integral Chapter 2. Lebesgue Measure on the Line Chapter 3. Integration on Measure Spaces Chapter 4. LP Spaces Chapter 5. The Caratheodory Construction of Measures Chapter 6. Product Measures Chapter 7. Lebesgue Measure on W1 and on Manifolds Chapter 8. Signed Measures and Complex Measures Chapter 9. LP Spaces, II Chapter 10. Sobolev Spaces Chapter 11. Maximal Functions and A.E. Phenomena Chapter 12. HausdorfTs r-Dimensional Measures Chapter 13. Radon Measures Chapter 14. Ergodic Theory VI Contents Chapter 15. Probability Spaces and Random Variables 207 Chapter 16. Wiener Measure and Brownian Motion 221 Chapter 17. Conditional Expectation and Martingales 233 Appendix A. Metric Spaces, Topological Spaces, and Compactness 251 Appendix B. Derivatives, Diffeomorphisms, and Manifolds 267 Appendix C. The Whitney Extension Theorem 277 Appendix D. The Marcinkiewicz Interpolation Theorem 283 Appendix E. Sard's Theorem 287 Appendix F. A Change of Variable Theorem for Many-to-one Maps 289 Appendix G. Integration of Differential Forms 293 Appendix H. Change of Variables Revisited 303 Appendix I. The Gauss-Green Formula on Lipschitz Domains 309 Bibliography 311 Symbol Index 315 Subject Index 317 vm Introduction Dominated Convergence Theorem. We integrate measurable functions de• fined on general measure spaces. Though at this point we have only con• structed Lebesgue measure on R, the basic theory of integration is not more complicated on general measure spaces, and pursuing it helps clarify what one should do to construct more general measures. In Chapter 4 we in• troduce LP spaces, consisting of measurable functions / such that |/|p is integrable or, more precisely, of equivalence classes of such functions, where we say f\ ~ fi provided these functions differ only on a set of measure zero. If /i is a measure on a space X, we study LP{X, fi) as a Banach space, for 1 < p < oo, and in particular we study L2(X, ji) as a Hilbert space. We develop some Hilbert space theory and apply it to establish the Radon- Nikodym Theorem, comparing two measures \i and v when v is "absolutely continuous" with respect to \i. Constructing measures other than Lebesgue measure on R is an impor• tant part of measure theory, and we begin this task in Chapter 5, giving some useful general methods, especially due to Caratheodory, for making such constructions, establishing that they are measures, and identifying cer• tain types of sets as measurable. The first concrete application of this is made in Chapter 6, in the construction of the "product measure" on X x Y, when X has a measure // and Y has the measure v. The integral with re• spect to the product measure /i x v is compared to "iterated integrals," in theorems of Fubini and Tonelli. In Chapter 7 we construct Lebesgue measure on Rn, for n > 1, as a product measure. We study how the Lebesgue integral on Rn transforms under an invertible linear transformation on W1 and, more generally, under a C1 diffeomorphism. We go a bit further, considering transformation via a Lipschitz homeomorphism, and we establish a result under the hypothe• sis that the transformation is differentiable almost everywhere, a property that will be studied further in Chapter 11. We extend the scope of the n- dimensional integral in another direction, constructing surface measure on an n-dimensional surface M in Rm. This is done in terms of the Riemann metric tensor induced on M. We go further and discuss integration on more general Riemannian manifolds. This central chapter contains a larger num• ber of exercises than the others, divided into several sets of exercises. After the first set, of a nature parallel to exercise sets for other chapters, there is a set relating the Riemann and Lebesgue integrals on Rn, extending the pre• vious discussion of the relationship between the material of Chapter 1 and that of Chapters 2-3, an exercise set on determinants, and an exercise set on row reduction on matrix products, providing linear algebra background for the proof of the change of variable formula. There is also an exercise set Introduction IX on the connectivity of G/(n,R) and a set on integration on certain matrix groups. In Chapter 8 we discuss "signed measures," which differ from the mea• sures considered up to that point only in that they can take negative as well as positive values. The key result established there is the Hahn de• composition, v — v^ — v~, for a signed measure v, on X, where v+ and v~ are positive measures, with disjoint supports. This allows us to extend the Radon-Nikodym Theorem to the case of a signed measure v, absolutely continuous with respect to a (positive) measure /i. We consider a further extension, to complex measures, though this is completely routine. In Chapter 9 we again take up the study of LP spaces and pursue it a bit further. We identify the dual of the Banach space LP(X, //) with Lq(X, //), where 1 < p < oo and 1/p + 1/q = 1, making use of the Radon-Nikodym Theorem for signed measures as a tool in the demonstration. We study some integral operators, including convolution operators, amongst others, and derive operator bounds on L9 spaces. We consider the Fourier transform T and prove that T is an invertible norm-preserving operator on L2(Rn). Some mention of Fourier series was made in Chapter 4, particularly in the exercises. Chapter 10 discusses Sobolev spaces, HkiP(Wri), consisting of functions whose derivatives of order < fc, defined in a suitable weak sense, belong to L^R72). Certain Sobolev spaces are shown to consist entirely of bounded continuous functions, even Holder continuous, on Rn. This subject is of great use in the study of partial differential equations, though such applications are not made here. The most significant application we make of Sobolev space theory in these notes appears in the following chapter. We mention that in Chapter 10 certain results on the weak derivative depend on the Fundamental Theorem of Calculus for the Riemann integral; it is for this reason that we included a proof of this result in Chapter 1. Chapter 11 deals with various results in the area of almost-everywhere convergence. The basic result, called Lebesgue's differentiation theorem, is that a function / G L1(Rn) is equal for almost all x G Rn to the limit of its averages over balls of radius r, centered at x, as r —> 0. Under a slightly stronger condition on x, we say x is a Lebesgue point for /; more generally there is the notion of an I^-Lebesgue point, and one shows that if / G Lp(Rn), then almost every x G Rn is an I^-Lebesgue point for /, provided 1 < p < oo. We use the Hardy-Littlewood maximal function as a tool to establish these results. This area also requires a "covering lemma," allowing one to select from a collection of sets covering S a subcollection with desirable properties. X Introduction Another important result established in Chapter 11 is Rademacher's Theorem that a Lipschitz function on M.n is differentiable almost everywhere. We get this as a corollary of the stronger result that if / G H1,p(M.n) and p > n (or p = n = 1), then / is differentiate almost everywhere; in fact, / is differentiate at every Lp-Lebesgue point of the weak derivative V/. With this result, we can complete the demonstration of the result from Chapter 7 that the change of variable formula for the integral extends to Lipschitz homeomorphisms. Making use of Rademacher's Theorem, we show that a Lipschitz function can be altered off a small set to yield a C1 function.