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## Über dieses Buch

Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and the existence of liftings.

Measure Theory provides a solid background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review of essential background material.

## Inhaltsverzeichnis

### Chapter 1. Measures

Abstract
In the most common construction of the Lebesgue integral of a function, the definition of the integral assumes that one knows the sizes of subsets of the function's domain. In Chapter 1 we introduce measures, the basic tool for dealing with such sizes. The first two sections of the chapter are abstract (but elementary). Section 1.1 looks at sigma-algebras, the collections of sets whose sizes we measure, while Section 1.2 introduces measures themselves. The heart of the chapter is in the following two sections, where we look at some general techniques for constructing measures (Section 1.3) and at the basic properties of Lebesgue measure (Section 1.4). The chapter ends with Sections 1.5 and 1.6, which introduce some additional fundamental techniques for handling measures and sigma-algebras.
Donald L. Cohn

### Chapter 2. Functions and Integrals

Abstract
Chapter 2 is devoted to the definition and basic properties of the Lebesgue integral. functions, the functions that are simple enough that the integral can be defined for them, if their values are not too large (Section 2.1). After a brief look in Section 2.2 at properties that hold almost everywhere (that is, that may fail on some set of measure zero, as long as they hold everywhere else), we turn to the definition of the Lebesgue integral and to its basic properties (Sections 2.3 and 2.4). The chapter ends with a sketch of how the Lebesgue integral relates to the Riemann integral (Section 2.5) and then with a few more details about measurable functions (Section 2.6).
Donald L. Cohn

### Chapter 3. Convergence

Abstract
In Chapter 3 we look in some detail at the convergence of sequences of functions. In Section 3.1 we define convergence in measure and convergence in mean, and we compare those modes of convergence with pointwise and almost everywhere convergence. In Section 3.2 we recall the definitions of norms and seminorms on vector spaces, and in Sections 3.3 and 3.4 we apply these concepts to the study of vector spaces of functions with integrable pth powers and to convergence in these spaces. Finally, in Section 3.5 we begin to look at dual spaces (the spaces of continuous linear functionals on normed vector spaces). We will continue the study of dual spaces in Sections 4.5, 7.3, and 7.5, by which time we will have developed enough tools to analyze and characterize a number of standard dual spaces.
Donald L. Cohn

### Chapter 4. Signed and Complex Measures

Abstract
In Chapter 4 we study signed and complex measures, which are like measures, but whose values may be either extended real numbers or complex numbers. We begin in Section 4.1 with some basic definitions and facts. Section 4.2 is devoted to the main result of this chapter, the Radon-Nikodym theorem, which characterizes those positive, signed, or complex measures whose values can be computed by integrating an integrable function. The last part of the chapter is devoted to the relation of the material in the early parts of the chapter to the classical concepts of bounded variation and absolute continuity (Section 4.4) and to the use of the Radon-Nikodym theorem to compute the dual spaces of a number of the vector spaces defined in Section 3.3 (Section 4.5).
Donald L. Cohn

### Chapter 5. Product Measures

Abstract
In calculus courses one defines integrals over two- (or higher-) dimensional regions and then evaluates these integrals by applying the usual techniques of integration, one variable at a time. In Chapter 5 we show that similar techniques work for the Lebesgue integral. More generally, given sigma-finite measures on two sets, we first define a natural product measure on the product of these sets (Section 5.1). Then we look at how integrals with respect to this product measure can be evaluated in terms of iterated integrals (Section 5.2). The chapter ends with a few applications (Section 5.3).
Donald L. Cohn

### Chapter 6. Differentiation

Abstract
In Chapter 6 we look at two aspects of the relationship between differentiation and integration. First, in Section 6.1, we look at changes of variables in integrals on Euclidean spaces. Such changes of variables occur, for example, when one evaluates an integral over a region in the plane by converting to polar coordinates. Then, in Sections 6.2 and 6.3, we look at some deeper aspects of differentiation theory, including the almost everywhere differentiability of monotone functions and of indefinite integrals and the relationship between Radon-Nikodym derivatives and differentiation theory. The Vitali covering theorem is an important tool for this. The discussion of differentiation theory will be resumed when we discuss the Henstock-Kurzweil integral in Appendix H.
Donald L. Cohn

### Chapter 7. Measures on Locally Compact Spaces

Abstract
Chapter 7 is devoted to the Riesz representation theorem and related results. The first section (Section 7.1) contains some basic facts about locally compact Hausdorff spaces, the spaces that provide the natural setting for the Riesz representation theorem, while the second section (Section 7.2) gives a proof of the Riesz representation theorem. The next two sections (Sections 7.3 and 7.4) contain some useful and relatively basic related material. The results of Sections 7.5 and 7.6 are needed for dealing with large locally compact Hausdorff spaces; for relatively small locally compact Hausdorff spaces (those that have a countable base), very few of the results in those two sections are needed.
The Daniell-Stone integral gives another way to deal with integration on locally compact Hausdorff spaces. Section 7.7 contains a result due to Kindler that summarizes the relationship of the Daniell-Stone integral to measure theory. The general Daniell-Stone setup is outlined in the exercises at the end of Section 7.7.
Donald L. Cohn

### Chapter 8. Polish Spaces and Analytic Sets

Abstract
The Borel subsets of a complete separable metric space have a number of interesting and useful characteristics. For example, two uncountable Borel subsets of complete separable metric spaces are necessarily Borel isomorphic, in the sense that there is a Borel measurable bijection from one to the other whose inverse is also Borel measuable. A related result says that if we have a Borel measurable injection from one complete separable metric space to another, then the image under this map of each Borel subset of the domain is Borel. If the function involved here is Borel but not necessarily injective, then the images of Borel sets are measurable for every finite Borel measure on the range space.
This Chapter is devoted to proving such results and to showing the context in which they arise.
Donald L. Cohn

### Chapter 9. Haar Measure

Abstract
Chapter 1 contains a proof that Lebesgue measure is translation invariant. It turns out that very similar results hold for every locally compact group (see Section 9.1 for the definition of such groups); the role of Lebesgue measure is played by what is called Haar measure. Chapter 9 is devoted to an introduction to Haar measure.
Section 9.1 contains some basic definitions and facts about topological groups. Section 9.2 contains a proof of the existence and uniqueness of Haar measure, and Section 9.3 contains additional basic properties of Haar measures. In Section 9.4 we construct two algebras that are fundamental for the study of harmonic analysis on a locally compact group.
Donald L. Cohn

### Chapter 10. Probability

Abstract
Chapter 10 is devoted to an introduction to probability theory. It contains some of the fundamental results of probability theory, in particular, the strong law of large numbers, the central limit theorem, the martingale convergence theorem, a construction of Brownian motion processes, and Kolmogorov’s consistency theorem.
Donald L. Cohn

### Backmatter

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