Lebesgue Integration

J.H. Williamson

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486789774
• eBook ISBN: 9780486796734
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

This concise introduction to Lebesgue integration is geared toward advanced undergraduate math majors and may be read by any student possessing some familiarity with real variable theory and elementary calculus. The self-contained treatment features exercises at the end of each chapter that range from simple to difficult.
The approach begins with sets and functions and advances to Lebesgue measure, including considerations of measurable sets, sets of measure zero, and Borel sets and nonmeasurable sets. A two-part exploration of the integral covers measurable functions, convergence theorems, convergence in mean, Fourier theory, and other topics. A chapter on calculus examines change of variables, differentiation of integrals, and integration of derivatives and by parts. The text concludes with a consideration of more general measures, including absolute continuity and convolution products.

[ 1 ]

Sets and Functions

1.1. Generalities

We take the usual simple-minded analyst’s view of set theory. The terms set, collection, aggregate, class, and so forth, are looked on as the same and will not be defined. The elements that make up a set may be called points, irrespective of their nature (often they will be points, in the usual sense, in Euclidean space). A collection of sets will be called a class rather than a set. We shall tend to use lower-case italic letters for points, italic capitals for sets, and script capitals for classes.

As usual, the property that the element a is in the set A (a belongs to A, A contains a) is denoted by a ∈ A. The negation of this is written a ∉ A. If a ∈ A implies a ∈ B (every element in A is also in B), then A is a subset of B, and we write A ⊂ B or B ⊃ A. If A ⊂ B and B ⊂ A, then the sets A and B are equal; A = B. If A ⊂ B and A ∉ B, then A is a proper subset of B. If A consists of the finite number of points a, b, · · ·, k, we write A = {a, b, · · ·, k}. More generally, if A consists of all points a for which the statement P(a) is true, we write A = {a: P(a)}.

It is convenient to introduce the empty set, denoted by Ø, which contains no points; it is a subset of every set. Also, all sets under consideration at any time will be subsets of some “large” set, the universal set, whole space · · ·. This will usually be left to be understood from the context; it will almost always be Euclidean space of the appropriate dimension, or a suitable subset of this.

A map or mapping f of A into B, or a function f from A to B, is a correspondence which assigns to each a ∈ A an element f(a) ∈ B; f(a) is the image of a under f. The set A is the domain (of definition) of f; in order to specify a function completely, its domain should be given, but often we will leave this to be understood from the context; no confusion should result. If E is a subset of A, the set f (E) = {f(a): a ∈ E} is the image of E under f. The set f (A) is the range of f. If F ⊂ B, the subset of A defined by {a: f (a) ∈ F} is called the inverse image of F under f, and it is denoted by f⁻¹ (F). If f (A) = B, then f is a map on to B. If f (a) = f {a′) implies a = a′, then f is one to one (1-1).

Let A be a set. A family in A is a set I (the index set) and a mapping f of I into A. The element f (i) may be written a_i, and the family may be denoted by (a_i)_i∈I or simply (a_i). The most familiar example, of course, is where I is the set of integers from 1 to n, or the set of all positive integers; the family is then a finite or infinite sequence. A family in...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Chapter 1. Sets and Functions
• Chapter 2. Lebesgue Measure
• Chapter 3. The Integral I
• Chapter 4. The Integral II
• Chapter 5. Calculus
• Chapter 6. More General Measures
• Index