Measurable function

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