measurable function

A measurable function is function between the sets underlying two measurable spaces which preserves the structure of the spaces; specifically, the preimage of a measurable set is measurable.

Formally,
let $(X, Σ)$ and $(Y, T)$ be measurable spaces.
A function $f : X \to Y$ is measurable if for every $E \in T$ , then $f^{- 1} (E) \in Σ$ :

{x \in X : f (x) \in E} \in Σ