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.

Also see

• Results about outer measures can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 6$