Show that for a martingale $M$ and every $0 \leq r < s$ and every bounded $\mathscr{F}_r$ measurable variable $Z$, $N_t = Z(M_{s \wedge t} - M_{r \wedge t})$ is a martingale.
I want to compute $E[N_t \mid \mathscr{F}_u]$, and for $u \geq r$ there is no problem, but I don't know what to say for $u < r$. It's clear that the value of the martingale should be zero for $u < r$, but since $Z$ is not necessarily $\mathscr{F}_u$ measurable I cannot remove it from the conditional expectation as I would like.
According to Jean-Francois Le Gall (this is part of a proof in his book, Brownian Motion, Martingales, and Stochastic Calculus), the proof of this fact is "easy".
Let $u < r$. We have
\begin{align*} E[N_t \vert \mathscr{F}_u] & = E[Z(M_{s \wedge t} -M_{r \wedge t}) \vert \mathscr{F}_u] \\ &= E[ZE[M_{s \wedge t} -M_{r \wedge t} \vert \mathscr{F}_r] \vert \mathscr{F}_u] \\ &= 0. \end{align*}
Note however that $N_u = 0$, so $E[N_t \vert \mathscr{F}_u] = N_u$, as desired.