\begin{definition}[Definition:Measurable Set/Subsets of Real Space]
A subset $S$ of $\R^n$ is said to be '''Lebesgue measurable''', frequently just '''measurable''', {{iff}} 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 $\mathfrak M_{\R^n}$.
\end{definition}