Definition:Countably Additive Function

Definition
Let $\mathcal A$ be a $\sigma$-algebra.

Let $f: \mathcal A \to \overline {\R}$ be a real-valued function where $\overline {\R}$ denotes the set of extended real numbers.

Then $f$ is defined as countably additive iff:
 * $\displaystyle f \left({\bigcup_{i \ge 1} A_i}\right) = \sum_{i \ge 1} f \left({A_i}\right)$

where $\left \langle {A_i} \right \rangle$ is any sequence of pairwise disjoint elements of $\mathcal A$.

That is, for any countably infinite set of pairwise disjoint elements of $\mathcal A$, $f$ of their union equals the sum of $f$ of the individual elements.

This is also known as a sigma-additive function, also written $\sigma$-additive function.