Definition:Measurable Mapping
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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.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.1$