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:
 * $$f \left({\bigcup_{i \ge 1} A_i}\right) = \sum_{i \ge 1} f \left({A_i}\right)$$

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.

Also see
If $$f$$ is countably additive, it is also finitely additive.