Characterization of a martingale by bounded continuous-time stopped expectations ($\mathbb{E}{X}_T=\mathbb{E}{X}_0$)

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 $U=s{1}_A+t{1}_{A^c}$, where $s<t$ and $A\in {\mathcal{F}}_s$. The resulting identity $\mathbb{E}[(X_t-X_s)1_A]=0$ 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.