This is a lemma from Rogers and Williams' Diffusions, Markov Processes, and Martingales.
Here, $H$ is a uniformly bounded simple process defined by an infinite sum. Defining $$H \bullet M(t) = \sum Z_{i-1}\{M(T_i \wedge t) - M(T_{i-1}\wedge t)\}$$ as in (25.1), how can we show that $H\bullet M$ is in $\mathcal{M}_0^2$, which is the space of $L^2$ bounded martingales null at $0$?
I am not sure how we can deal with the infinite sum here to show martingale property and $L^2$ boundedness, in particular, how we get $E(H\bullet M)^2_\infty \le c^2 E(M_\infty^2)$. I would greatly appreciate any help.
