\begin{definition}[Definition:Pre-Measure]
Let $X$ be a set.
Let $\SS \subseteq \powerset X$ be a collection of subsets of $X$.
Let $\O \in \SS$.
Let $\mu: \SS \to \overline \R$ be a mapping, where $\overline \R$ denotes the set of extended real numbers.
Then $\mu$ is said to be a '''pre-measure''' {{iff}} it satisfies the following conditions:
:$(1): \quad$ For all $A \in \SS$, if $\map \mu A$ is finite then $\map \mu A \ge 0$.
:$(2): \quad \map \mu \O = 0$
:$(3): \quad$ For every sequence $\sequence {A_n}_{n \mathop \in \N}$ of pairwise disjoint sets in $\SS$ with $\ds \bigcup_{n \mathop \in \N} A_n \in \SS$:
::$\ds \map \mu {\bigcup_{n \mathop \in \N} A_n} = \sum_{n \mathop \in \N} \map \mu {A_n}$
:that is, that $\mu$ is countably additive.
\end{definition}