Real Analysis

Modern Techniques and Their Applications

Gerald B. Folland

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118626399
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

An in-depth look at real analysis and its applications-now expanded and revised. This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory. This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.

Chapter 1

Measures

In this chapter we set forth the basic concepts of measure theory, develop a general procedure for constructing nontrivial examples of measures, and apply this procedure to construct measures on the real line.

1.1 INTRODUCTION

One of the most venerable problems in geometry is to determine the area or volume of a region in the plane or in 3-space. The techniques of integral calculus provide a satisfactory solution to this problem for regions that are bounded by “nice” curves or surfaces but are inadequate to handle more complicated sets, even in dimension one. Ideally, for n

we would like to have a function μ that assigns to each E ⊂

ⁿ a number μ(E)

[0, ∞], the n-dimensional measure of E, such that μ(E) is given by the usual integral formulas when the latter apply. Such a function μ should surely possess the following properties:

i. If E₁, E₂,… is a finite or infinite sequence of disjoint sets, then

ii. If E is congruent to F (that is, if E can be transformed into F by translations, rotations, and reflections), then μ(E) = μ(F).

iii. μ(Q) = 1, where Q is the unit cube

Unfortunately, these conditions are mutually inconsistent. Let us see why this is true for n = 1. (The argument can easily be adapted to higher dimensions.) To begin with, we define an equivalence relation on [0, 1) by declaring that x ~ y iff x – y is rational. Let N be a subset of [0, 1) that contains precisely one member of each equivalence class. (To find such an N, one must invoke the axiom of choice.) Ne...