Let $X_t$ be a martingale with a filtration $\{F\}_t$. Prove that $X_t 1_{r(X_t)}$ is a martingale where $r(X_t)$ is some relation involving $X_t$, for example $ X_t \leq 0 $.
My attempt
We know $\mathbb{E}[X_t 1_{A}] = \mathbb{E}[X_m 1_{A}]$ for $\forall A \in F_m $ and $t > m $. Note that $1_{r(X_t)}$ is $F_t$ measurable while $1_{A}$ is $F_m$ measurable. Hence the indicator of their intersection is $F_m$ measurable. Therefore $\mathbb{E}[X_t 1_{r(X_t)} 1_{A} ] = \mathbb{E}[X_t 1_{A \cap r(X_t)}] = \mathbb{E}[X_m 1_{A \cap r(X_m)}] = \mathbb{E}[X_m 1_{A} 1_{ r(X_m)}]$ I am not completely sure about the second equality of the proof where my indicator of ${X_t}$ becomes an indicator of $X_m$. Is this correct ? Any help would be greatly appreciated.