Definition:Outer Measure

From ProofWiki
Jump to: navigation, search

Definition

Let $X$ be a set, and let $\mathcal P \left({X}\right)$ be its power set.

An outer measure (on $X$) is a mapping:

$\mu^*: \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$

that satisfies the following conditions:

$(1): \quad \mu^* \left({\varnothing}\right) = 0$
$(2): \quad \mu^* \left({A}\right) \le \mu^* \left({B}\right)$ for all $A, B \in \mathcal P \left({X}\right)$ with $A \subseteq B$ (that is, $\mu^*$ is monotone)
$(3): \quad \displaystyle \mu^* \left({\bigcup_{i \mathop \in \N} A_i}\right) \le \sum_{i \mathop = 1}^\infty \mu^* \left({A_i}\right)$ for all sequences $\left({A_i}\right)_{i \mathop \in \N} \in \mathcal P \left({X}\right)$ (that is, $\mu^*$ is countably subadditive)

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


Sources