Definition:Induced Outer Measure

Definition
Let $\mu$ be a pre-measure on a semiring $\mathcal S$ over a set $X$.

The outer measure induced by the pre-measure $\mu$ is defined as:


 * $\displaystyle \mu^* \left({S}\right) = \inf \ \left\{ {\sum_{n=1}^\infty \mu \left({A_n}\right) : \forall n \in \N: A_n \in \mathcal S, \ S \subseteq \bigcup_{n=1}^\infty A_n} \right\}$

Here, the infimum is taken in the extended real numbers.