A Signal Theoretic Introduction to Random Processes

2 BACKGROUND: MATHEMATICS

2.1 INTRODUCTION

This chapter details relevant mathematical theory, and mathematical results, to provide the basis for analysis in subsequent chapters and to ensure a coherent and systematic development of material. A summary of mathematical notation is detailed in the notation table in the reference section at the end of the book.

A basic introduction to set theory, and function theory, is provided, and this is followed by an introduction to measure theory. Two types of measure are considered. The first, Lebesgue measure, underpins Lebesgue integration, and this is fundamental to signal theory, which is discussed in Chapter 3. The second, Lebesgue–Stieltjes measure, provides the basis for Lebesgue–Stieltjes integration. Lebesgue–Stieltjes integration underpins Cramer’s representation of a signal, which is discussed in Chapter 3. The discussion, in general, complements that of Howard (2002, ch. 2).

2.2 SET THEORY

2.2.1 Basic Definitions

Definition: Set

A set is a collection of distinct entities.

The notation {α₁, α₂, …, α_N} is used for the set of distinct entities α₁, α₂, …, α_N. The notation {x : f(x)} is used for the set of elements x for which the property f(x) is true. The notation means that the entity, denoted x, is an element of the set S.

The empty set is denoted by . The complement of a set S, denoted S^C, is defined as where ...