Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 2

MEASURES

Measurable sets are to measure theory as open sets are to topology (Williams, 1991). Measures are set functions defined on measurable sets. These concepts are used later to define integration. In this chapter, we shall introduce measurable sets, measures, and other relevant concepts such as algebras and σ-algebras.

2.1 Basic Concepts and Facts

Definition 2.1 (Algebra). An algebra or field ∑₀ on S is a collection of subsets of S that satisfies the following conditions:

(a) S ∑₀.

(b) If F ∑₀, then F^c ∑₀, where F^c = S\F.

(c) If F ∑₀ and G ∑₀, then F ∪ G ∑₀.

Definition 2.2 (σ-Algebra). A collection ∑ of subsets of S is called a σ-algebra or σ-field if ∑ is an algebra on S and is closed under countably infinite unions; that is, if F_n ∑ for n = 1, 2,…, then

Definition ...