See Measure theory

A measurable function between two sets $X$ and $Y$, belonging to Measurable spaces $(X, A)$, and $(Y, B)$, is {a Function $f: X \rightarrow Y$, s.t. for any $E \in B$, the Preimage of $E$ is in $A$}. I.e. the preimage of any set in the Sigma-algebra of the co-domain is in the Sigma-algebra of the domain.

https://en.wikipedia.org/wiki/Measurable_function