Definition:Countably Subadditive Function

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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 iff, for any sequence $\left \langle {E_n} \right \rangle_{n \in \N}$ of elements of $\Sigma$:

$\displaystyle f \left({\bigcup_{n \mathop = 0}^\infty E_n}\right) \le \sum_{n \mathop = 0}^\infty f \left({E_n}\right)$


Also known as

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