Definition:Pre-Image Sigma-Algebra/Domain

Definition

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:

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

It is a $\sigma$-algebra, as proved on Pre-Image Sigma-Algebra on Domain 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 ... (previous) ... (next): $3.3 \ \text{(vii)}$