Definition:Measurable Mapping

Definition

Let $\left({X, \Sigma}\right)$ and $\left({X', \Sigma'}\right)$ be measurable spaces.

A mapping $f: X \to X'$ is said to be $\Sigma \, / \, \Sigma'$-measurable iff:

$\forall E' \in \Sigma': f^{-1} \left({E'}\right) \in \Sigma$

That is, iff the preimage of every measurable set under $f$ is again measurable.

Also known as

When no danger of ambiguity arises, a $\Sigma \, / \, \Sigma'$-measurable mapping $f$ may simply be called measurable.