Definition:Outer Measure

Definition
Let $X$ be a set.

Let $\powerset X$ be the power set of $X$.

An outer measure (on $X$) is a mapping:
 * $\mu^*: \powerset X \to \overline \R_{\ge 0}$

that satisfies the following conditions:


 * $(1): \quad \map {\mu^*} \O = 0$
 * $(2): \quad \forall A, B \in \powerset X: A \subseteq B \implies \map {\mu^*} A \le \map {\mu^*} B$ (that is, $\mu^*$ is monotone)
 * $(3): \quad \ds \map {\mu^*} {\bigcup_{i \mathop = 1}^\infty A_i} \le \sum_{i \mathop = 1}^\infty \map {\mu^*} {A_i}$ for all sequences $\sequence {A_i}_{i \mathop \in \N} \in \powerset X$ (that is, $\mu^*$ is countably subadditive)

where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.