Definition:Measure (Measure Theory)

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


 * $$\mu \left({S}\right) \ge 0$$ for each $$S \in A$$;
 * $$\mu \left({\bigcup_{n=1}^{\infty} S_n}\right) = \sum_{n=1}^{\infty} \mu \left({S_{n}}\right)$$ for every sequence of pairwise disjoint sets $$\left\{{S_{n}}\right\} \subseteq A$$ (that is, $$\mu\ $$ is a countably additive function).