# Definition:Measurable Set

## Contents

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

## Also see

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

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.1$