Definition:Pre-Image Sigma-Algebra/Codomain

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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:

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

It is a $\sigma$-algebra, as proved on Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra.

Also known as

As usual, one may also write pre-image sigma-algebra.