# Definition:Countably Additive Function

## Definition

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:

- $\displaystyle f \left({\bigcup_{n \mathop \in \N} E_n}\right) = \sum_{n \mathop \in \N} f \left({E_n}\right)$

where $\left \langle {E_n} \right \rangle$ 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.

## Also known as

This is also known as a **sigma-additive function** or a **$\sigma$-additive function**.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.11$: Problems: $17$