Definition:Countably Subadditive Function

Definition

Let $\Sigma$ be a $\sigma$-algebra over a set $X$.

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 subadditive if and only if for any sequence $\sequence {E_n}_{n \mathop \in \N}$ of elements of $\Sigma$:

$\ds \map f {\bigcup_{n \mathop = 0}^\infty E_n} \le \sum_{n \mathop = 0}^\infty \map f {E_n}$

Also known as

A countably subadditive function is also known as a sigma-subadditive function or $\sigma$-subadditive function.