Let $X$ be a càdlàg adapted integrable process. The following are equivalent: 1. $X$ is a martingale: $X_t \in L^1$ for every $t$, and $\mathbb{E}[X_t \mid \mathcal{F}_s] = X_s$ for all $t > s$. 2. The stopped process $X^T = (X_{T \wedge t})_{t \geq 0}$ is a martingale for every stopping time $T$. 3. For all stopping times $T, S$ with $T$ bounded, $X_T \in L^1$ and $\mathbb{E}[X_T \mid \mathcal{F}_S] = X_{T \wedge S}$ almost surely. 4. For all bounded stopping times $T$, $X_T \in L^1$ and $\mathbb{E}[X_T] = \mathbb{E}[X_0]$. For $X$ uniformly integrable, conditions (3) and (4) hold for all stopping times, not just bounded ones.