\begin{definition}[Definition:Countably Additive Function]
Let $\Sigma$ be a $\sigma$-algebra.
Let $f: \Sigma \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers.
Then $f$ is defined as '''countably additive''' {{iff}}:
:$\ds \map f {\bigcup_{n \mathop \in \N} E_n} = \sum_{n \mathop \in \N} \map f {E_n}$
where $\sequence {E_n}$ is any sequence of pairwise disjoint elements of $\Sigma$.
That is, for any countably infinite set of pairwise disjoint elements of $\Sigma$, $f$ of their union equals the sum of $f$ of the individual elements.
\end{definition}