Real Analysis: Measures, Integrals and Applications (Universitext)
Anatolii Podkorytov
Language: English
Pages: 772
ISBN: 1447151216
Format: PDF / Kindle (mobi) / ePub
Real Analysis: Measures, Integrals and Applications is devoted to the basics of integration theory and its related topics. The main emphasis is made on the properties of the Lebesgue integral and various applications both classical and those rarely covered in literature.
This book provides a detailed introduction to Lebesgue measure and integration as well as the classical results concerning integrals of multivariable functions. It examines the concept of the Hausdorff measure, the properties of the area on smooth and Lipschitz surfaces, the divergence formula, and Laplace's method for finding the asymptotic behavior of integrals. The general theory is then applied to harmonic analysis, geometry, and topology. Preliminaries are provided on probability theory, including the study of the Rademacher functions as a sequence of independent random variables.
The book contains more than 600 examples and exercises. The reader who has mastered the first third of the book will be able to study other areas of mathematics that use integration, such as probability theory, statistics, functional analysis, partial probability theory, statistics, functional analysis, partial differential equations and others.
Real Analysis: Measures, Integrals and Applications is intended for advanced undergraduate and graduate students in mathematics and physics. It assumes that the reader is familiar with basic linear algebra and differential calculus of functions of several variables.
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the Dirichlet problem is unique. To outline an approach that can lead to finding the solution, assume that the closure is a standard compact set. If U is a solution of the Dirichlet problem that is smooth in some neighborhood of , then, according to the integral representation formula (see Theorem 8.7.3), for every , one has (6) The right-hand side of this formula contains an unknown function . To eliminate it, we will do the following. Fix a point and consider a function W x harmonic on whose
references to the appropriate literature are provided for the interested reader. The notion of surface area is discussed in more detail than is common in analysis texts. Using a descriptive definition, we prove its uniqueness on Borel subsets of smooth and Lipschitz manifolds. It is desirable that the reader be familiar with the notion of an integral of a continuous function of one variable on an interval prior to being exposed to the basics of measure theory. However, we do not feel that this
invariant under rigid motions (see Corollary 2.4.4), the Lebesgue measure on a subspace does not depend on the motion used in its construction. It also follows immediately from the definition that the Lebesgue measures in subspaces transform into each other under rigid motions; in this sense, they form a coherent family. The Lebesgue measure on a k-dimensional affine subspace will be denoted by the same symbol λ k as the measure on . It will always be clear from the context on which subspace the
introduce the notion of measure and establish its basic properties. The next two sections are devoted to the extension of a measure by the Carathéodory method and to properties of such an extension. Theorem 1.5.1 on the uniqueness of an extension proved here is repeatedly used in the book. In Sect. 1.6, we study properties of the Borel hull of a system of sets. 1.1 Systems of Sets In classical analysis, one usually works with functions that depend on one or several numerical variables, but
j x k . 7. For which values of a and b is the integral finite, where m⩾2? Express it in terms of the beta function. 8. Prove that, for every non-negative measurable function f on and all , the relation holds. 9. Using the previous problem and induction, prove that, for p∈(−1,1) and , the relation where , holds. 10. Regarding the plane as the set of complex numbers, find a function ω>0 such that the measure ν with density ω>0 (dν=ω dλ 2) is invariant under multiplication, i.e., such that
