Definition:Induced Outer Measure

Definition
Let $\mathcal S$ be a collection of subsets of a set $X$, and suppose that $\varnothing \in \mathcal S$.

Let $\mu$ be a pre-measure on $\mathcal S$.

The outer measure induced by the pre-measure $\mu$ is the mapping $\mu^* : \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ defined as:


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

Here, $\mathcal P \left({X}\right)$ 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.

Also see

 * Induced Outer Measure Restricted to Semiring is Pre-Measure
 * Elements of Semiring are Measurable with Respect to Induced Outer Measure
 * Carathéodory's Theorem (Measure Theory)