Suppose $(\Omega,\mathcal F,\{\mathcal F_n\}_n,\mathbb P)$ is a filtered probability space. Let $\tau$ be a stopping time, and define a collection of sets $\mathcal F_\tau:=\{A\in\mathcal F: A\cap\{\tau=n\}\in\mathcal F_n\forall n<\infty\}$. We can easily show that $\mathcal F_\tau$ is a $\sigma$-algebra.
Let $M=\{M_n\}_n$ be a martingale. Show that $\mathbb E[M_n|\mathcal F_\tau]=M_{n\wedge\tau}, n\ge 0$.
Let $M=\{M_n\}_n$ be a uniformly integrable martingale with last element $M_\infty\in\mathcal F_\infty$. Show that $\mathbb E[M_\infty|\mathcal F_\tau]=M_\tau$.
My attempt:
I would use the definition of conditional expectation to show both of the identities.
For any $S\in\mathcal F_\tau$, we want to show that $\mathbb E[M_{n\wedge\tau};S]=\mathbb E[M_n;S]$. But then I got completely stuck for hours. Even after checking a lot of posts dealing with this kind of problem, I am still quite confused with the definition of $\mathcal F_\tau$. Can someone help me out? Thank you!
First we have to show that $M_{n\wedge \tau}$ is $\mathcal F_{\tau}$ measurable. For this we have to show that $(M_{n\wedge \tau} \leq x)\cap (\tau=k) \in \mathcal F_k$. But $(M_{n\wedge \tau} \leq x)\cap (\tau=k)=(M_{n\wedge k} \leq x)\cap (\tau=k)$ which is an intersection of two sets in $\mathcal F_k$.
Next we have to show that $\int_A M_{n\wedge \tau} dP=\int_A M_n dP$ for $A \in \mathcal F_{\tau}$. It is enough to show that $\int_{A\cap (\tau=k)} M_{n\wedge \tau} dP=\int_{A\cap (\tau=k)} M_n dP$ for each $k$. In other words we have to show $\int_{A\cap (\tau=k)} M_{n\wedge k} dP=\int_{A\cap (\tau=k)} M_n dP$. This follows by martingale property if $n > k$ and trivially holds if $n \leq k$.
The second part is just obtained by taking limit as $n \to \infty$ since a uniformly integrable martingale converges almost surely and in $L^{1}$.