Real Variables PDF

Author’s Foreword

During the academic year 1985-1986 I gave a course on Real Variables at Indiana University. The main source of reference for the course was a set of class notes prepared by the students as we went along; this book is based on those notes. One of the purposes in those lectures was to present to students who are beginning a deeper study of the fairly esoteric subject of Real Variables an overview of how the familiar results covered in Advanced Calculus develop into a rich theory. Motivation is an essential ingredient in this endeavour, as are convincing examples and interesting applications.

Now, teaching a course at this level two facts become quickly apparent, to wit: (i) The background of the students is quite varied, as first year graduate and upper division undergraduate Math students, as well as various science and economics majors, enroll in it, and, (ii) Even those students with a strong background are not entirely at ease with proofs involving either an abstract new concept or an e-6 argument. My idea of a course at this level is one that presents to the students a modern introduction to the theory of real variables without subjecting them to undue stress.

Although the material is not presented here in a radically different way than in other textbooks, this book offers a conceptually different approach. First, it takes into account, both in placement and content of the topics discussed, the uneven nature of the background of the students. Second, an attempt has been made to motivate the material discussed, and always the most “natural” rather than the most elegant proof of a result is given. We also stress the unity of the subject matter rather than individual results. Third, we go from the particular to the general, discussing each definition and result rather carefully, closer to the way a mathematician first thinks about a new concept. Finally, students are not “talked down,” but rather feel that the issues at hand are addressed in a forthright manner and in a direct language, one they can understand. It is important that readers have no difficulty in following the actual arguments presented and spend their time instead in considering questions such as: What is the role or roles of a given result? What is it good for? What are the important ideas, and which are the secondary ones? What are the basic problems in this area and how are they approached and solved?

In fact, we expect the serious students at this level to learn to ask these questions and this text will serve as a guide to ask them at the appropriate time.

How does the text present the material? An important consideration is that the students see the “ big picture” rather than isolated theorems, and basic ideas rather than generality are stressed. Each chapter starts with a short reader’s guide stating the goals of the chapter. Specific examples are discussed, and general concepts are developed through particular cases. There are 646 problems and questions that are used to motivate the material as well as to round out the development of the subject matter. The reader will be pleasantly surprised to find out that problems are in fact problems, and not further theorems to be proved. The problems are thought-provoking, and there is a mixture of routine to difficult, and concrete to theoretical.

Because I wanted this book to be essentially self-contained for those students with a good Advanced Calculus background as well as an elementary knowledge of the theory o f metric spaces, the point of departure is an informal discussion of the theory of sets and cardinal numbers in Chapter I, and ordinal numbers and Zorn’s Lemma in Chapter n. These topics give the student the opportunity to work with abstract, possible new, concepts. Chapter III introduces the Riemann-Stieltjes integral and the limitations of the Riemann integral become quickly apparent; e-6 proofs are discussed here. At the completion of these chapters the background of the students has been essentially equalized. Chapter IV is the exception that proves the rule. It develops the abstract concept of measure, a particular case of which, the Lebesgue measure on Rⁿ, is discussed in Chapter V. Anyone objecting to this treatment can plainly, and almost painlessly, read these chapters in the opposite order. The construction of the Lebesgue measure is a favorite among the students, as it allows them to discover where measures come from and how they are constructed.

In Chapter VI we return to a somewhat abstract setting, although for reasons of simplicity Lusin’s theorem is presented in the line where all the difficulties are already apparent. An important feature of this chapter is working with “good” and “bad” sets; this is an indispensable tool in other areas, including the Calderon-Zygmund decomposition of integrable functions discussed in Chapter VIII. The proof of Egorov’s theorem illustrates our point of view: It is longer than the usual proof, but it is clear and understandable. In Chapter VII we introduce the notion of the integral and the role o f almost everywhere convergence. I am confident that the path that leads to the various convergence theorems is direct and motivational. The material described thus far constitutes a solid first semester of a yearlong course.

Chapter VIII presents new properties of integrable functions, including the Lebesgue Differentiation Theorem. The proof given here makes use of the Hardy-Littlewood maximal function, and is one that most experts agree should have worked its way into the standard treatment of this topic by now. Chapter IX constructs important new examples of measures on the line, the Borel measures. The correspondence between these measures and their distribution functions, a subject that lies at the heart of the theory of Probability, is established in an elementary and computational manner. Chapter X discusses properties of absolutely continuous functions, including the Lebesgue decomposition of functions of bounded variation and the characterization of those functions on the line that may be recovered by integrating their derivatives. The abstract setting of these results is presented in detail in Chapter XI, where the Radon-Nikodym theorem is discussed. The basic theory of the Lebesgue L^p spaces, including duality and the notion of weak convergence is covered in Chapter XII. Chapter XIII deals with product measures and Fubini’s theorem in the following manner: In the first section we discuss the version dealing with Lebesgue integrals in Euclidean space; the second section discusses some important applications, including convolutions and approximate identities; and, finally, the third section presents Fubini’s theorem in an abstract setting. This is a concrete example on how to proceed from the particular to the general. However, if preferred, the third and second sections can be covered, and the first section assigned for reading.

Chapter XIV deals with normed linear spaces, an abstraction of the notion of the L^p spaces, and the Hahn-Banach theorems. Students are happy to see both the geometric and analytic forms of this result and their applications. Chapter XV covers the basic principles of Functional Analysis, to wit, the Uniform Boundedness Principle, the Closed Graph Theorem, and the Open Mapping Theorem; each principle is given individual attention. In Chapter XVI we consider those Banach spaces whose norm comes from an inner product, or Hilbert spaces. The discussion of the geometry o f Hilbert spaces and the spectral decomposition of compact self-adjoint operators are some of the features of this chapter.

Brief historical references concerning the origin of some of the concepts introduced in the text have been made throughout the text, and Chapter XVII presents these remarks in their natural setting, namely, the theory of Fourier series. Finally, Chapter XVIII contains suggestions and comments to some of the problems and questions posed in the book; they are not meant, however, to make the learning of the material effortless.

The notations used throughout the book are either standard or else they are explained as they are introduced. “Theorem 3.2” means that the result alluded to appears as the second item in Section 3 of the present chapter, and “Theorem 3.2 in Chapter X” means that it appears as the second item of the third section in Chapter X. The same convention is used for formulas and problems.

A word about where the text fits into the existing literature. It is more advanced than Rudin’s book Principles of Mathematical Analysis, a good reference for the material on Advanced Calculus and metric spaces. It is also more abstract than the treatise Measure and Integral by Wheeden and Zygmund. I learned much of the material on integration from Antoni Zygmund, and some of the topics discussed, including the construction of the Lebesgue measure and the outlook on the Euclidean version of Fubini’s theorem, have his imprint. Then, there are the classics. They include Natanson’s Theory of Functions of a Real Variable, Saks’ Theory of the Integral, F.Riesz and Sz.-Nagy’s Legons d\Analyse Fonctionnelle, Halmos’ Measure Theory, Hewitt and Stromberg’s Real and Abstract Analysis, and Dunford and Schwartz’s Linear Operators. Anyone consulting these books will gain the perspective of the masters.

Where do we go from here? I am confident that the reading of this book will adequately prepare the student to venture into diverse fields of Mathematics. Specifically, books such as Billingsley’s Probability and Measure, Conway’s A Course in Functional Analysis, Stein’s Singular Integrals and Differentiabilty Properties of Functions and Zygmund’s Trigonometric Series are now within reach

Author: Alberto Torchinsky