Definition:Measurable Mapping

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Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces.

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

$\forall E' \in \Sigma': f^{-1} \sqbrk {E'} \in \Sigma$

That is, if and only if 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.