\begin{definition}[Definition:Countably Subadditive Function]
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 $\sequence {E_n}_{n \mathop \in \N}$ of elements of $\Sigma$:
:$\ds \map f {\bigcup_{n \mathop = 0}^\infty E_n} \le \sum_{n \mathop = 0}^\infty \map f {E_n}$
\end{definition}