I've been finding some problems with this exercise on stopping times.
Let X a random variable on $(\Omega,\mathcal{F},\{\mathcal{F_t}\}_{t\in T},\mathbb{P})$ filtered space and let $\tau$ a stopping time. Prove that $X$ is $\mathcal{F}_\tau$-measurable if and only if $X\textbf{1}_{\{\tau=n\}}$ is $\mathcal{F}_n$-measurable for all $n$.
Edit:
($\Rightarrow$) assume that $X$ is $\mathcal{F}_\tau$-measurable and that takes value on the space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, then we have that events of the form $\{X\in A\}$ for some $A\in\mathcal{B}(\mathbb{R})$ satisfies $\{X\in A\}\cap\{\tau\leq n\}\in\mathcal{F}_n$ for all $n$.Then we want to prove that the r.v. $X\textbf{1}_{\tau=n}:(\Omega,\mathcal{F}_n)\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $B\in \mathcal{B}(\mathbb{R})$ we have that: \begin{equation} \{X\textbf{1}_{\{\tau=n\}}\in B\}= \begin{cases} \{X\in B\}\cap\{\tau=n\} \ \ \ \text{if} \ 0\notin B\\ (\{X\in B\}\cap\{\tau=n\})\cup(\{X=0\}\cup\{\tau\neq n\}) \ \ \ \text{if} \ 0\in B\\ \end{cases} \end{equation} Now in the first case we have that since by hyothesis $\{X\in B\}\in\mathcal{F}_\tau$ we have that $\{X\in B\}\cap\{\tau\leq n\}\in\mathcal{F}_n$ and then: \begin{equation} \{X\in B\}\cap\{\tau=n\}=\{X\in B\}\cap\{\tau\leq n\}\cap\{\tau\leq n-1\}^c\in\mathcal{F}_n \end{equation} In the second case we have just to prove that $\{X=0\}\cup\{\tau\neq n\}\in\mathcal{F}_n$. First it is easy to see that $\{X=0\}\in\mathcal{F}_\tau$, then: \begin{equation} \{X=0\}\cup\{\tau\neq n\}=\{X=0\}\cup\{\tau=n\}^c=\{\{X=0\}^c\cap\{\tau=n\}\}^c \end{equation} Where the last $\in\mathcal{F}_n$ using same argument of before we conclude.
($\Leftarrow$) Assume now that $X\textbf{1}_{\tau=n}$ is $\mathcal{F}_n$-measurable. Then let $A\in\mathcal{B}(\mathbb{R})$ we want to check that $\{X\in A\}\in\mathcal{F}_\tau$. Thus we have to prove that: \begin{equation} \{X\in A\}\cap\{\tau\leq n\}\in\mathcal{F}_n \end{equation} Now we have that: $\{X\in A\}=\bigcup_{k\in\mathbb{N}}\{X\in A\}\cap\{\tau=k\}$ where all the events $\{X\in A\}\cap\{\tau=k\}\in\mathcal{F}_k$. Now we conclude: \begin{equation} \begin{split} \{X\in A\}\cap\{\tau\leq n\}&=\bigg\{\bigcup_{k\in\mathbb{N}}\{X\in A\}\cap\{\tau=k\}\bigg\}\cap\{\tau\leq n\}\\ &=\bigcup_{k\leq n}\{X\in A\}\cap\{\tau=k\} \in\mathcal{F}_n \end{split} \end{equation}
I've tried to compute the proof but I'm not completely sure on the first implication, can someone help me?