Definition:Pre-Image 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:

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


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