Definition:Trace Sigma-Algebra

Definition
Let $X$ be a set, and let $\mathcal A$ be a $\sigma$-algebra on $X$.

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

Then the trace $\sigma$-algebra (of $E$ in $\mathcal A$), $\mathcal{A}_E$, is defined as:


 * $\mathcal{A}_E := E \cap \mathcal A = \left\{{E \cap A: A \in \mathcal A}\right\}$

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 $E$).