Definition:Measurable Set/Subset of Reals

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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 $A \subseteq \R$:

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

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

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