Outer Measure and Measure

A function = (A) that is defined for every subset A of a set S is called an outer measure if it satisfies the following:

(i) (A) ≥ 0, (∅) = 0. (ii) (A1) ≤ (A2) if A1 ⊂ A2. (iii) (

⋃

Ak) ≤ ∑

(Ak) for any countable collection of sets {Ak}.