= Characterization of a martingale by bounded continuous-time stopped expectations
{title2=$\mathbb E X_T=\mathbb E X_0$}
An integrable <adapted process> is a <martingale> if all its values at <bounded stopping times> are integrable with the same <expectation> as $X_0$. For the converse, compare deterministic $t$ with $U=s\mathbf1_A+t\mathbf1_{A^c}$, where $s<t$ and $A\in\mathcal F_s$. The resulting identity $\mathbb E[(X_t-X_s)\mathbf1_A]=0$ is precisely the defining test for <conditional expectation>. In the forward direction, use the <optional stopping theorem> for a <càdlàg martingale> under the <usual conditions for a filtration>.
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