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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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) equation

where ak [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

equation

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

(6.2) equation

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

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