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Über dieses Buch

This first year graduate text is a comprehensive resource in real analysis based on a modern treatment of measure and integration. Presented in a definitive and self-contained manner, it features a natural progression of concepts from simple to difficult. Several innovative topics are featured, including differentiation of measures, elements of Functional Analysis, the Riesz Representation Theorem, Schwartz distributions, the area formula, Sobolev functions and applications to harmonic functions. Together, the selection of topics forms a sound foundation in real analysis that is particularly suited to students going on to further study in partial differential equations.

This second edition of Modern Real Analysis contains many substantial improvements, including the addition of problems for practicing techniques, and an entirely new section devoted to the relationship between Lebesgue and improper integrals. Aimed at graduate students with an understanding of advanced calculus, the text will also appeal to more experienced mathematicians as a useful reference.

Inhaltsverzeichnis

Chapter 1. Preliminaries

Abstract
This is the first of three sections devoted to basic definitions, notation, and terminology used throughout this book. We begin with an elementary and intuitive discussion of sets and deliberately avoid a rigorous treatment of “set theory” that would take us too far from our main purpose.
William P. Ziemer

Chapter 2. Real, Cardinal, and Ordinal Numbers

Abstract
A brief development of the construction of the real numbers is given in terms of equivalence classes of Cauchy sequences of rational numbers. This construction is based on the assumption that properties of the rational numbers, including the integers, are known.
William P. Ziemer

Chapter 3. Elements of Topology

Abstract
The purpose of this short chapter is to provide enough point set topology for the development of the subsequent material in real analysis. An in-depth treatment is not intended. In this section, we begin with basic concepts and properties of topological spaces.
William P. Ziemer

Chapter 4. Measure Theory

Abstract
An outer measure on an abstract set X is a monotone, countably subadditive function defined on all subsets of X. In this section, the notion of measurable set is introduced, and it is shown that the class of measurable sets forms a $$\sigma$$-algebra, i.e., measurable sets are closed under the operations of complementation and countable unions. It is also shown that an outer measure is countably additive on disjoint measurable sets.
William P. Ziemer

Chapter 5. Measurable Functions

Abstract
The class of measurable functions will play a critical role in the theory of integration. It is shown that this class remains closed under the usual elementary operations, although special care must be taken in the case of composition of functions. The main results of this chapter are the theorems of Egorov and Lusin. Roughly, they state that pointwise convergence of a sequence of measurable functions is “nearly” uniform convergence and that a measurable function is “nearly” continuous.
William P. Ziemer

Chapter 6. Integration

Abstract
Based on the ideas of H. Lebesgue, a far-reaching generalization of Riemann integration has been developed. In this section we define and deduce the elementary properties of integration with respect to an abstract measure.
William P. Ziemer

Chapter 7. Differentiation

Abstract
Certain covering theorems, such as the Vitali covering theorem, will be developed in this section. These covering theorems are of essential importance in the theory of differentiation of measures.
William P. Ziemer

Chapter 8. Elements of Functional Analysis

Abstract
We have already encountered examples of normed linear spaces, namely the $$L^{p}$$ spaces. Here we introduce the notion of abstract normed linear spaces and begin the investigation of the structure of such spaces.
William P. Ziemer

Chapter 9. Measures and Linear Functionals

Abstract
Theorem 6.​48 states that a function in $$L^{p^\prime }$$ can be regarded as a bounded linear functional on $$L^{p}$$. Here we show that a large class of measures can be represented as bounded linear functionals on the space of continuous functions. This is a very important result that has many useful applications and provides a fundamental connection between measure theory and functional analysis.
William P. Ziemer

Chapter 10. Distributions

Abstract
In the previous chapter, we saw how a bounded linear functional on the space $$C_{c}(\mathbb {R}^{n})$$ can be identified with a measure. In this chapter, we will pursue this idea further by considering linear functionals on a smaller space, thus producing objects called distributions that are more general than measures. Distributions are of fundamental importance in many areas such as partial differential equations, the calculus of variations, and harmonic analysis. Their importance was formally acknowledged by the mathematics community when Laurent Schwartz, who initiated and developed the theory of distributions, was awarded the Fields Medal at the 1950 International Congress of Mathematicians. In the next chapter, some applications of distributions will be given. In particular, it will be shown how distributions are used to obtain a solution to a fundamental problem in partial differential equations, namely, the Dirichlet problem. We begin by introducing the space of functions on which distributions are defined.
William P. Ziemer

Chapter 11. Functions of Several Variables

Abstract
Because of the central role played by absolutely continuous functions and functions of bounded variation in the development of the fundamental theorem of calculus in $$\mathbb {R}$$, it is natural to ask whether they have analogues among functions of more than one variable. One of the main objectives of this chapter is to show that this is true. We have found that the BV functions in $$\mathbb {R}$$ constitute a large class of functions that are differentiable almost everywhere. Although there are functions on $$\mathbb {R}^{n}$$ that are analogous to BV functions, they are not differentiable almost everywhere. Among the functions that are often encountered in applications, Lipschitz functions on $$\mathbb {R}^{n}$$ form the largest class that are differentiable almost everywhere. This result is due to Rademacher and is one of the fundamental results in the analysis of functions of several variables.
William P. Ziemer

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