Definition:Pre-Image Sigma-Algebra

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:

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

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:

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

Also see

• Pre-Image Sigma-Algebra on Domain is Sigma-Algebra
• Pre-Image Sigma-Algebra on Codomain is Sigma-Algebra
• Pre-Image Sigma-Algebra on Domain makes Mapping Measurable