Let $\mu$ be a [[Set Function|set function]]. The function is finitely additive if $\mu(A \cup B) = \mu(A)+\mu(B)$ for all disjoint sets $A,\,B$ in its domain We can prove by mathematical induction that $\mu$ is *countably additive*, i.e. $\mu\left( \bigcup_{j=1}^{\infty} A_{j} \right) = \sum_{j=1}^{\infty} \mu(A_{j})$for every $A_{j}$ disjoint sets in the domain.