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

CHAPTER 6

LEBESGUE INTEGRATION

Lebesgue integration is the general theory of integration of a function with respect to a general measure. In this chapter, we present some concepts and theorems used to develop the Lebesgue integration theory.

6.1 Basic Concepts and Facts

Definition 6.1 (Simple Function). Let (S, ∑) be a measurable space and (m∑)⁺ denote the set of all nonnegative ∑-measurable functions. An element f of (m∑)⁺ is considered simple if f can be written as a finite sum

(6.1)

where a_k [0, ∞] and _k ∑ for k = 1, 2,…, m. The set of all nonnegative simple functions is denoted by SF⁺.

Definition 6.2 (Integral of Nonnegative Simple Functions). Let (S, ∑, μ) be a measure space, and let

be a nonnegative simple function on S. The integral of f with respect to μ is defined as

(6.2)

Definition 6.3 (Integral of Nonnegative Measurable Functions). Let (S, ∑, μ) be a measure space and (m