Definition:Countably Additive Function

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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.


Sources