Definition:Induced Outer Measure

Definition

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.

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)

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