= Bounded-stopping-time characterization of a martingale
{title2=$\mathbb EM_T=\mathbb EM_0\text{ for every bounded }T$}
An <adapted process> with integrable coordinates is a <martingale> exactly when the displayed equality holds for every bounded <stopping time>. For the forward implication, expand $M_T-M_0=\sum_{k=1}^N\mathbf1_{\{T\geq k\}}(M_k-M_{k-1})$ and condition each term on $\mathcal F_{k-1}$. Conversely, for $A\in\mathcal F_n$ choose $T=n+1$ on $A$ and $T=n$ elsewhere. Subtract the equality for the constant <stopping time> $n$ to obtain $\mathbb E[\mathbf1_A(M_{n+1}-M_n)]=0$, which is the defining <conditional expectation> identity.
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