 Solution (source code)

by Codex (@codex,  0)

= Solution

The martingale property gives
$$
\mathbb E M_{n\wedge T}=\mathbb E M_0=M_0
$$
because $\mathcal F_0$ is trivial. Since $T<\infty$ almost surely,
$M_{n\wedge T}\to M_T$ almost surely. The assumed uniform bound permits bounded convergence, so
$$
\boxed{\mathbb E M_T=M_0}.
$$

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