Definition:Pre-Image Sigma-Algebra
(Redirected from Definition:Preimage Sigma-Algebra)
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Definition
On Domain
Let $X, X'$ be sets.
Let $f: X \to X'$ be a mapping.
Let $\Sigma'$ be a $\sigma$-algebra on $X'$.
Then the pre-image $\sigma$-algebra (of $\Sigma'$) on the domain of $f$ is defined as:
- $f^{-1} \sqbrk {\Sigma'} := \set {\map {f^{-1} } {E'}: E' \in \Sigma'}$
On Codomain
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping.
Let $\Sigma$ be a $\sigma$-algebra on $X$.
Then the pre-image $\sigma$-algebra (of $\Sigma$) on the codomain of $f$ is defined as:
- $\Sigma' := \left\{{E' \subseteq X': f^{-1} \left({E'}\right) \in \Sigma}\right\}$