Let $s < t$. Let $\tau$ be a stopping time. Let M, N square integrable martingale.
In a proof that i'm following i read that: $$E\big[E[N_{\tau}(M_{\tau} - M_t)\mathbb1_{\tau\le t}|\mathcal F_{s\lor\tau}]|\mathcal F_s\big]= E[N_{\tau}(M_{\tau} - M_{\tau\lor s})\mathbb1_{\tau\le t}|\mathcal F_s]$$
what i don't understand is why $\mathbb1_{\tau\le t}$ can go outside the expectation without have any change, it seems that the explanation is that it is $F_{s\lor\tau}$ measurable, but instead i think that it is $F_{t}$ measurable and that $ F_{s\lor\tau} \subset F_{t}$. Any help?
Indeed, $\{\tau\leq t\}\in F_{s\lor\tau}$. By definition, $\sigma$-algebra $\mathcal F_\tau$ is $$ \mathcal F_\tau:=\left\{A\in\mathcal F:A\cap\left\{\tau\le u\right\}\in\mathcal F_u\;\text{for all }u\right\} $$ Then $$ F_{s\lor\tau} = \{A\in\mathcal F: A\cap \{\max(s,\tau) \leq u\} \in \mathcal F_u \text{ for all } u\} = \{A\in\mathcal F: A\cap \{s\leq u\}\cap \{\tau \leq u\} \in \mathcal F_u \text{ for all } u\} $$ Look at $A=\{\tau\leq t\}$ and check whether it belongs to $F_{s\lor\tau}$. $$ \{\tau\leq t\}\cap\{s\leq u\}\cap \{\tau \leq u\} = \{\tau\leq t\land u\}\cap\{s\leq u\} \in \mathcal F_u. $$ So, $\{\tau\leq t\}\in F_{s\lor\tau}$.