Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $\left(\mathcal{F}_n, n \in \mathbb N\right)$ a filtration. Let $T$ be a stopping time with respect to $\left(\mathcal{F}_n, n \in \mathbb{N}\right)$. One defines the information one possesses at time $T$ as the following $\sigma$-field: $$ \mathcal{F}_T := \left\{A \in \mathcal{F}: A \cap\{T=n\} \in \mathcal{F}_n, \forall n \in \mathbb{N}\right\}. $$
I would like to verify below result, i.e.,
Theorem A random variable $Y$ is $\mathcal F_T$-measurable IFF $Y1_{\{T=n\}}$ is $\mathcal F_n$-measurable for all $n \in \mathbb N$.
Could you have a check on my below attempt?
Proof Fix $B \in \mathcal B(\mathbb R)$.
Let $Y$ be $\mathcal F_T$-measurable. We want to prove $\{Y1_{\{T=n\}} \in B\} \in \mathcal F_n$. Indeed, $$ \{Y1_{\{T=n\}} \in B\} = (\{Y\in B\} \cap \{T=n\}) \cup (\{T \neq n\} 1_{\{0 \in B\}}) \in \mathcal F_n. $$
Let $Y1_{\{T=n\}}$ be $\mathcal F_n$-measurable for all $n \in \mathbb N$. We want to prove $\{Y \in B\} \cap \{T=n\}$ for all $n \in \mathbb N$. Indeed, $$ \{Y \in B\} \cap \{T=n\} = \{Y1_{\{T=n\}} \in B\} \setminus (\{T \neq n\} 1_{\{0 \in B\}}) \in \mathcal F_n. $$