Definition:Trace Sigma-Algebra

This page is about trace $\sigma$-algebras. For other uses, see Trace.

Definition

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$.

Let $E \subseteq X$ be a subset of $X$.

Then the trace $\sigma$-algebra (of $E$ in $\Sigma$), $\Sigma_E$, is defined as:

$\Sigma_E := \set {E \cap S: S \in \Sigma}$

It is a $\sigma$-algebra on $E$, as proved on Trace $\sigma$-Algebra is $\sigma$-Algebra.

Also known as

The trace $\sigma$-algebra may also be called the trace sigma-algebra, the induced $\sigma$-algebra (on $E$) or the induced sigma-algebra (on $E$).

It is common to write $E \cap \Sigma$ for $\Sigma_E$, but this can cause confusion; hence it is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Results about trace $\sigma$-algebras can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $3.3 \ \text{(vi)}$