Measurable function

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