Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space with filtration $(\mathcal F_t)_{t\in T}$, and let $(X_t)_{t\in T}$ be an integrable adapted process. Then $(X_t)_{t\in T}$ is a martingale if and only if for every $s\le t$,
\begin{align*}
\mathbb E[X_t-X_s\mid\mathcal F_s]=0
\end{align*}
$\mathbb P$-a.s.