Definition:Measure (Measure Theory)

A measure on a $\sigma\ $-algebra $$A\ $$ is a function $$\mu:A \to \R$$ such that


 * $$\mu(S) \geq 0$$ for each $$S\in A$$, and
 * $$\mu\Big(\bigcup_{n=1}^{\infty}S_n\Big) = \sum_{n=1}^{\infty}\mu (S_{n})$$ for every sequence of pairwise disjoint sets $$\{S_{n}\}\subseteq A$$. (I.e., $$\mu\ $$ is a countably additive function.)