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 if and only if:

$\ds \map f {\bigcup_{n \mathop \in \N} E_n} = \sum_{n \mathop \in \N} \map f {E_n}$

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.

Also known as

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

Also see

• Results about countably additive functions can be found here.

Sources

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