Definition:Measurable Set/Arbitrary Outer Measure

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Definition

Let $\struct {X, \Sigma}$ be a measurable space.


Let $\mu^*$ be an outer measure on $X$.

A subset $S \subseteq X$ is called $\mu^*$-measurable if and only if it satisfies the Carathéodory condition:

$\map {\mu^*} A = \map {\mu^*} {A \cap S} + \map {\mu^*} {A \setminus S}$

for every $A \subseteq X$.


By Set Difference as Intersection with Complement, this is equivalent to:

$\map {\mu^*} A = \map {\mu^*} {A \cap S} + \map {\mu^*} {A \cap \map \complement S}$

where $\map \complement S$ denotes the relative complement of $S$ in $X$.

The collection of $\mu^*$-measurable sets is denoted $\map {\mathfrak M} {\mu^*}$ and is a $\sigma$-algebra over $X$.


Sources


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