Definition:Measurable Set/Subsets of Real Space

Definition
A subset $S$ of $\R^n$ is said to be Lebesgue measurable, frequently just measurable, 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 $\mathfrak M_{\R^n}$.