# 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**.