Definition:Measurable Mapping

Definition

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.

Also see

• Results about measurable mappings can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): measurable function
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.1$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): measurable function