# 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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- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.1$ - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**measurable set**