Measure Theory

Frontmatter

1. Measures

Abstract

Let X be an arbitrary set. A collection A of subsets of X is an algebra on X if

(a)

X ∈ A,

 

(b)

for each set A that belongs to A the set A ^c belongs to A,

 

(c)

for each finite sequence A ₁,..., A _n of sets that belong to A the set \( \cup _{i = 1}^n {\rm A}_i \) belongs to A, and

 

(d)

for each finite sequence A ₁,..., A _n of sets that belong to A the set \( \cap _{i = 1}^n {\rm A}_i \) belongs to A.

 

Donald L. Cohn

2. Functions and Integrals

Abstract

In this section we introduce measurable functions and study some of their basic properties. We begin with the following elementary result.

Donald L. Cohn

3. Convergence

Abstract

In this section we define and study a few modes of convergence for sequences of measurable functions. For simplicity we shall discuss only real-valued functions. It should be easy to see that everything can be extended so as to apply to complex-valued functions and to [−∞,+∞]-valued functions that are finite almost everywhere.*

Donald L. Cohn

4. Signed and Complex Measures

Abstract

Let (X, A) be a measurable space, and let μ be a function on A with values in [−∞,+∞]. The function μ is finitely additive if the identity

$$\mu \left( {\bigcup\limits_{i = 1}^n {{\rm A}_i } } \right) = \sum\limits_{i = 1}^n {\mu ({\rm A}_i )} $$

Donald L. Cohn

5. Product Measures

Abstract

This chapter is devoted to measures and integrals on product spaces. The first two sections contain the basic facts about product measures and about the evaluation of integrals on product spaces; the last section contains some applications.

Donald L. Cohn

6. Differentiation

Abstract

In this section we deal with changes of variable in R ^d and with their relation to Lebesgue measure. The main result is Theorem 6.1.6. Let us begin by recalling some definitions.

Donald L. Cohn

7. Measures on Locally Compact Spaces

Abstract

In this chapter we shall be dealing with measures and integrals on locally compact Hausdorff spaces. This first section contains a summary of some of the necessary topological facts and constructions; the main development begins in Section 2.

Donald L. Cohn

8. Polish Spaces and Analytic Sets

Abstract

A Polish space is a separable topological space that can be metrized by means of a complete metric. This section contains a number of elementary properties of Polish spaces. In Sections 3 through 6 below we shall use these properties, plus the concept of an analytic set (see Section 2), to derive some deep and useful results about measurable sets and functions.

Donald L. Cohn

9. Haar Measure

Abstract

A topological group is a set G that has the structure of a group (say with group operation (x,y) → xy) and of a topological space, and is such that the operations (x,y) → xy and x → x ⁻¹ are continuous. Note that (x,y) → xy is a function from the product space G × G to G, and that we are requiring that it be continuous with respect to the product topology on G × G; thus xy must be “jointly continuous” in x and y, and not merely continuous in x with y held fixed and continuous in y with x held fixed (see Exercise 3). A locally compact topological group, or simply a locally compact group, is a topological group whose topology is locally compact and Hausdorff. A compact group is a topological group whose topology is compact and Hausdorff.

Donald L. Cohn

Backmatter