Measurable Functions

Definition: Measurable Functions

Let $(X,{\Sigma }_X)$ and $(Y,{\Sigma }_Y)$ be measurable spaces and let $f:X\to Y$ be a function.

We say that $f$ is measurable if the inverse image $f^{-1}(E)$ of each $E\in {\Sigma }_Y$ is in ${\Sigma }_X$:

$$E\in {\Sigma }_Y⟹f^{-1}(E)\in {\Sigma }_X$$