Show that this set is a sigma algebra

Question

Let $(X, \mathcal A)$ be a measurable set, $Y$ a subset of $X$

Show that $\mathcal A_Y$$= { A \cap Y : A \in \mathcal A } $ is a $\sigma$-algebra on $Y$

Answer 1

We have $\mathcal{A}_Y=\{A\cap Y:A\in\mathcal{A}\}$. Let $B\in \mathcal{A}_Y$. Hence, there exists $A\in \mathcal{A}$ such that $B=A\cap Y$. Accordingly, $Y\setminus B=(X\setminus A)\cap Y\in \mathcal{A}_Y$, due to $X\setminus A\in \mathcal{A}_Y$. The other properties of a $\sigma$-algebra are easy to verify.