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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 if and only if 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.

Also see