Definition:Measurable Set

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

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

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:


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

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 $\R$ is said to be Lebesgue measurable, or frequently just measurable, 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 $\mathfrak M$.

Using the axiom of choice, it can be demonstrated that there exist non-measurable subsets of $\R$.

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


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

where $m^*$ is defined as:


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

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 $\complement \left({S}\right)$ is the complement of $S$ in $\R^n$.

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