Definition:Measurable Set

Definition
Let $X$ be a set, and let $\mathcal A$ be a $\sigma$-algebra on $X$.

A set $A \in \mathcal A$ is said to be ($\mathcal A$-)measurable.

Measurable Sets of an Arbitrary Outer Measure
Given an outer measure $\mu^*\ $ on a set $X\ $, a subset $S\subseteq X$ is called $\mu^*\ $-measurable if it satisfies the Carathéodory condition:


 * $\mu^*(A) = \mu^*(A\cap S) + \mu^*(A - S)$

for every $A\in\mathcal P(X)$.

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

Measurable Subsets of the Reals
A subset $E$ of the reals is said to be Lebesgue measurable, frequently just measurable, if for every set $A \in \R$:


 * $m^*A = m^*(A \cap E) + m^*(A \cap \mathcal C \left ({E}\right))$

where $m^*$ is defined as described in the definition of Lebesgue measure and $\mathcal C \left ({E}\right)$ is the complement of $E$ in $\R$.

The set of all measurable sets of $\R$ is frequently denoted $\mathfrak M_\R$ or just $\mathfrak M$.

There are sets in $\mathcal{P} \left({\R}\right)$ which are not in $\mathfrak M$.

Measurable Subsets of $\R^n$
A subset $E$ of $\R^n$ is said to be Lebesgue measurable, frequently just measurable, if for every set $A \in \R^n$:


 * $m^*A = m^*(A \cap E) + m^*(A \cap \mathcal C \left ({E}\right))$

where $m^*$ is defined as:


 * $\displaystyle m^*(E) = \inf_{\left\{{I_k}\right\} :E \subseteq \cup I_k} \sum v (I_k)$

where $\left\{{I_k}\right\}$ are a sequence of sets satisfying


 * $I_k = [a_1,b_1] \times \dots \times [a_n,b_n] $

In the definition, infimum ranges over all such sets $\left\{{I_n}\right\}$, and $v(I_n)$ is the "volume" $\displaystyle \prod_{i=1}^n |b_i-a_i|$, and $\mathcal C \left ({E}\right)$ is the complement of $E$ in $\R^n$.

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