Definition:Measurable Set

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Definition

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

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


Measurable Set of an Arbitrary Outer Measure

Let $\mu^*$ be an outer measure on $X$.

A subset $S \subseteq X$ is called $\mu^*$-measurable if and only if 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 Subset of the Reals

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 $A \in \R$:

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

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 Subset of $\R^n$

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 $\mathfrak M_{\R^n}$.


Also see


Sources


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