Measurable set

A measurable set in a measurable space $(X,\Sigma)$ is a subset $A\subseteq X$ with $A\in\Sigma$.

Measurable sets are precisely the subsets to which a measure assigns a value, and they determine measurable functions via preimages. The indicator function of a measurable set is a basic example of a measurable function.

Examples:

• In $(\mathbb R,\mathcal B(\mathbb R))$ with the Borel sigma-algebra , every open interval such as $(a,b)$ is measurable.
• If $A$ is measurable in $(X,\Sigma)$, then its complement $X\setminus A$ is also measurable.
• If $(A_n)_{n\ge 1}$ are measurable, then the countable union $\bigcup_{n=1}^\infty A_n$ is measurable.