Definition:Trace Sigma-Algebra

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.