Definition – Measure. – Math-reference.net

Contents
1. Definition
2. Fourth Condition

Definition

Let $X$ be a set and let $S \subseteq \mathcal{P}(X)$ . such that

$A\in S \Leftrightarrow A^\complement \in S$ .
For an infinite sequence of sets, $\{A_1, ..., A_n\}$ , where no restrictions are placed on the sets $\{A_1, ..., A_n\}$ other than that they are elements of $S$ , we have that,

$\bigcup\limits_{i=1}^n A_i \in S$ .

We refer to $S$ as a sigma algebra. A measure is a function $m: S \rightarrow \mathbb{R}$ such that, for any two disjoint sets $A_1, A_2 \in S$

$m(A_1\cup A_2) = m(A_1) + m(A_2)$ .

Properties of a measure

Via proof by induction, we can extend the measure condition $m(A_1\cup A_2) = m(A_1) + m(A_2)$ to any finite number of disjoint sets,

$m(A_1\cup ... \cup A_n) = m(A_1) + ... + m(A_n)$ .

2. Since $(A_1\setminus A_2)\cup A_2 = A_1 \cup A_2$ , we have that

$m((A_1\setminus A_2)\cup A_2) = m(A_1) + m(A_2) \\ = m(A_1\setminus A_2) + m(A_2)$ .