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.

**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 *A* ⊆ *B* ⊆ *S*, then μ*(*A*) ≤ μ* (*B*).

(c) (Countable subadditivity) If *A*_{n} ⊆ *S* for all *n* ≥ 1, then

**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 *B* ⊆ *A*.

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

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