Let $P = (\Omega, \mathcal{F}, \mathbb{P},\{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space.
A D5696: Complex random process $Z : J \to \mathsf{Random}(\Omega \to \mathbb{C})$ is a martingale on $P$ if and only if

(1)	$Z$ is a D2156: Adapted random collection on $P$
(2)	\begin{equation} \forall \, j \in J : \mathbb{E} |Z_j| < \infty \end{equation}
(3)	\begin{equation} \forall \, i, j \in J \left( i < j \quad \implies \quad \mathbb{E}(Z_j \mid \mathcal{G}_i) \overset{a.s.}{=} Z_i \right) \end{equation}