Measurable Set

measure-theory

Definition

Measurable Set

Let $(X,\mathfrak{A})$ be a measurable space. A set $A\subseteq X$ is measurable if it is an element of the sigma-algebra:

$$A\in \mathfrak{A}.$$

In a measure space, measurable sets are the sets to which a measure assigns a value.

Not every subset of $X$ is measurable; measurability depends on the chosen $\sigma$-algebra.