# Definition:Measurable Set/Subset of Reals

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## Definition

A subset $S$ of the real numbers $\R$ is said to be **Lebesgue measurable**, or frequently just **measurable**, if and only if for every set $A \in \R$:

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

where $\lambda^*$ is the Lebesgue outer measure.

The set of all **measurable sets** of $\R$ is frequently denoted $\mathfrak M_\R$ or just $\mathfrak M$.

## 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**