Outer-measure

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Definition

An outer-measure, $\mu^*$ is a set function from a hereditary $\sigma$-ring, $\mathcal{H}$, to the (positive) extended real values, $\bar{\mathbb{R} }_{\ge0}$, that is:

• $\forall A\in\mathcal{H}[\mu^*(A)\ge 0]$ - non-negative
• $\forall A,B\in\mathcal{H}[A\subseteq B\implies \mu^*(A)\le\mu^*(B)]$ - monotonic
• $\forall ({ A_n })_{ n = 1 }^{ \infty }\subseteq \mathcal{H} [\mu^*(\bigcup_{n=1}^\infty A_n)\le\sum^\infty_{n=1}\mu^*(A_n)]$ - countably subadditive

In words, $\mu^*$ is:

• an extended real valued countably subadditive set function that is monotonic and non-negative with the property: $\mu^*(\emptyset)=0$ defined on a hereditary $\sigma$-ring

For every pre-measure

$$\mu^*=\text{Inf}\left.\left\{\sum^\infty_{n=1}\bar{\mu(E_n)}\right\vert E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}$$ is an outer measure.