Definition:Measurable Set

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Definition

Let $\left({X, \Sigma}\right)$ be a measurable space.

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


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 if and only if it satisfies the Carathéodory condition:

$\mu^* \left({A}\right) = \mu^* \left({A \cap S}\right) + \mu^* \left({A \setminus S}\right)$

for every $A \subseteq X$.

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

$\mu^* \left({A}\right) = \mu^* \left({A \cap S}\right) + \mu^* \left({A \cap \complement \left({S}\right)}\right)$

where $\complement \left({S}\right)$ denotes the relative complement of $S$ in $X$.

The collection of $\mu^*$-measurable sets is denoted $\mathfrak M \left({\mu^*}\right)$ 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, if and only if for every set $A \in \R$:

$\lambda^* \left({A}\right) = \lambda^* \left({A \cap S}\right) + \lambda^* \left({A \setminus S}\right)$

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, if and only if for every set $A \in \R^n$:

$m^*A = m^* \left({A \cap S}\right) + m^* \left({A \cap \complement \left({S}\right)}\right)$

where:

$\complement \left({S}\right)$ is the complement of $S$ in $\R^n$
$m^*$ is defined as:
$\displaystyle m^* \left({S}\right) = \inf_{\left\{ {I_k}\right\}: S \mathop \subseteq \cup I_k} \sum v \left({I_k}\right)$

where:

$\left\{{I_k}\right\}$ are a sequence of sets satisfying:
$I_k = \left[{a_1 \,.\,.\, b_1}\right] \times \dots \times \left[{a_k \,.\,.\, b_k}\right]$



$v \left({I_n}\right)$ is the "volume" $\displaystyle \prod_{i \mathop = 1}^n \left\vert{b_i - a_i}\right\vert$
the infimum ranges over all such sets $\left\{{I_n}\right\}$


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


Also see


Sources