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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 3

EXTENSION OF MEASURES

As defined in the previous chapter, a measure is a set function on an algebra. To extend a measure from an algebra to a σ-algebra, we need some tools. In this chapter, we will introduce concepts and theorems related to measure extension.

3.1 Basic Concepts and Facts

Definition 3.1 (Outer Measure). An outer measure on a set is a nonnegative, extended real-valued set function that satisfies the following conditions:

(a) = 0.
(b) (Monotonicity) If ABS, then μ*(A) ≤ μ* (B).
(c) (Countable subadditivity) If AnS for all n ≥ 1, then

equation

Definition 3.2 (Complete Measure). Let μ be a measure on a σ-algebra ∑. The measure μ, is said to be complete if and only if all subsets of a zero measure set in ∑ are in ∑. Thus, whenever A ∑ with μ(A) = 0, we have B ∑ for all BA.

Definition 3.3 (Completion of Measure Spaces). Let (S, ∑, μ) be a measure space. The completion ...

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