Definition:Trace Sigma-Algebra

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

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


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 ProofWiki.


Sources