Definition:Measurable Set

Definition
Let $\struct {X, \Sigma}$ be a measurable space.

A subset $S \subseteq X$ is said to be ($\Sigma$-)measurable $S \in \Sigma$.

Measurable Sets of an Arbitrary Outer Measure
Let $\mu^*$ be an outer measure on a set $X$.

A subset $S \subseteq X$ is called $\mu^*$-measurable it satisfies the Carathéodory condition:


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

for every $A \subseteq X$.

By Set Difference as Intersection with Complement, this is equivalent to:


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

where $\map \complement S$ denotes the relative complement of $S$ in $X$.

The collection of $\mu^*$-measurable sets is denoted $\map {\mathfrak M} {\mu^*}$ and is a $\sigma$-algebra over $X$.

Measurable Subsets of the Reals
A subset $S$ of the real numbers $\R$ is said to be Lebesgue measurable, or frequently just measurable, 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 $\mathfrak M$.

Measurable Subsets of $\R^n$
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:
 * $\displaystyle \map {m^*} S = \inf_{\set {I_k}: S \mathop \subseteq \cup I_k} \sum \map v {I_k}$

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" $\displaystyle \prod_{i \mathop = 1}^n \size {b_i - a_i}$


 * the infimum ranges over all such sets $\set {I_n}$

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

Also see

 * Outer Measure Restricted to Measurable Sets is Measure


 * Existence of Non-Measurable Subset of Real Numbers: from the axiom of choice, it is demonstrated that there exist non-measurable subsets of $\R$.