= Solution
The <optional stopping theorem> for a right-continuous <martingale> $M$ on a <filtered probability space> satisfying the <usual conditions for a filtration> states that, for bounded <stopping times> $\sigma\leq\tau$,
$$
\boxed{\mathbb E[M_\tau\mid\mathcal F_\sigma]=M_\sigma,
\qquad \mathbb EM_\tau=\mathbb EM_\sigma.}
$$
For a <submartingale> the conditional equality becomes $\mathbb E[M_\tau\mid\mathcal F_\sigma]\geq M_\sigma$, and for a <supermartingale> it becomes the reverse inequality.
For a <uniformly integrable martingale>, boundedness of the <stopping times> can be removed, with the terminal limit used if a <stopping time> is infinite. A useful application is to a bounded <stopped martingale> $M_{t\wedge\tau}$ with $\tau<\infty$ <almost surely>: apply the bounded-time theorem first and then the <bounded convergence theorem> to obtain $\mathbb EM_\tau=\mathbb EM_0$. Finiteness of $\tau$ alone is not a sufficient hypothesis for an arbitrary <martingale>.
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