\begin{definition}[Definition:Induced Outer Measure]
Let $\SS$ be a collection of subsets of a set $X$, and suppose that $\O \in \SS$.
Let $\mu$ be a pre-measure on $\SS$.
The '''outer measure induced by the pre-measure $\mu$''' is the mapping $\mu^* : \powerset X \to \overline \R_{\ge 0}$ defined as:
:$\ds \map {\mu^*} S = \inf \set {\sum_{n \mathop = 1}^\infty \map \mu {A_n}: \forall n \in \N: A_n \in \SS, \ S \subseteq \bigcup_{n \mathop = 1}^\infty A_n}$
Here, $\powerset X$ denotes the power set of $X$, and $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
The infimum of the empty set is taken to be $+\infty$.
It follows immediately by Construction of Outer Measure that the induced outer measure is an outer measure.
\end{definition}