# Definition:Pre-Measure

## Definition

Let $X$ be a set.

Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Let $\varnothing \in \mathcal S$.

Let $\mu: \mathcal S \to \overline{\R}_{\ge 0}$ be a mapping, where $\overline{\R}_{\ge 0}$ denotes the set of positive extended real numbers.

Then $\mu$ is said to be a pre-measure iff it satisfies the following conditions:

$(1):\quad \mu \left({\varnothing}\right) = 0$
$(2):\quad$ For every sequence $\left({A_n}\right)_{n \in \N}$ of pairwise disjoint sets in $\mathcal S$ with $\displaystyle \bigcup_{n \mathop \in \N} A_n \in \mathcal S$:
$\displaystyle \mu \left({\bigcup_{n \mathop \in \N} A_n}\right) = \sum_{n \mathop \in \N} \mu \left({A_n}\right)$
that is, that $\mu$ is countably additive.