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.