Trace sigma-algebra

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Definition

Let $(X,\mathcal{A})$ be a $\sigma$-algebra and let $Y\subseteq X$ be any subset of $X$, then we may construct a $\sigma$-algebra on $Y$ called the trace $\sigma$-algebra, $\mathcal{A}_Y$ given by:

• $\mathcal{A}_Y:=\left\{Y\cap A\ \vert A\in\mathcal{A}\right\}$

Claim: $(Y,\mathcal{A}_Y)$ is a $\sigma$-algebra

Proof of claims

[Expand]

Claim 1: that $(Y,\mathcal{A}_Y)$ is indeed a $\sigma$-algebra

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Easy - just show definition