Definition:Pre-Image Sigma-Algebra/Codomain

Definition

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:

$\Sigma' := \set {E' \subseteq X': f^{-1} \sqbrk {E'} \in \Sigma}$

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.

Sometimes, this is called merely the pre-image $\sigma$-algebra, but this leads to confusion with either the pre-image $\sigma$-algebra on the domain or the pre-image $\sigma$-algebra on the codomain.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales: $7.2$: Proof