# 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

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

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