Measurable Function

measure-theory

Definition

Measurable Function

Let $(X,\mathfrak{A})$ and $(Y,\mathfrak{B})$ be measurable spaces. A function $f:X\to Y$ is measurable if the preimage of every measurable set is measurable:

$$\forall B\in \mathfrak{B},f^{-1}(B)\in \mathfrak{A}.$$

In measure theory, this condition ensures that a function can be integrated and studied using measure-theoretic methods.

The most common case is a function $f:X\to \mathbb{R}$ or $f:X\to \mathbb{C}$ on a measure space, where measurability is defined with respect to the usual $\sigma$-algebra on the codomain.