Definition:Pre-Measure

Definition
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 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

 * Measure, a refinement which imposes that $\mathcal S$ be a $\sigma$-algebra.