# Measurable function

In mathematics, a function f that maps each element of a measurable space ${\displaystyle \scriptstyle (X,{\mathcal {F}}_{X})}$ to an element of another measurable space ${\displaystyle \scriptstyle (Y,{\mathcal {F}}_{Y})}$ is said to be measurable (with respect to the sigma algebra ${\displaystyle \scriptstyle {\mathcal {F}}_{X}}$) if for any set ${\displaystyle \scriptstyle A\in {\mathcal {F}}_{Y}}$ it holds that ${\displaystyle \scriptstyle f^{-1}(A)\in {\mathcal {F}}_{X}}$, where ${\displaystyle \scriptstyle f^{-1}(A)=\{x\in X\mid f(x)\in A\}}$.