# Definition:Measurable Set/Subsets of Real Space

## Definition

A subset $S$ of $\R^n$ is said to be Lebesgue measurable, frequently just measurable, if and only if for every set $A \in \R^n$:

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

where:

$\map \complement S$ is the complement of $S$ in $\R^n$
$m^*$ is defined as:
$\ds \map {m^*} S = \inf_{\set {I_k}: S \mathop \subseteq \cup I_k} \sum \map v {I_k}$

where:

$\set {I_k}$ are a sequence of sets satisfying:
$I_k = \closedint {a_1} {b_1} \times \dots \times \closedint {a_k} {b_k}$
$\map v {I_n}$ is the volume $\ds \prod_{i \mathop = 1}^n \size {b_i - a_i}$
the infimum ranges over all such sets $\set {I_n}$

The set of all measurable sets of $\R^n$ is frequently denoted $\mathfrak M_{\R^n}$.