Let $P = (\Omega, \mathcal{F}, \mathbb{P},\{ \mathcal{G}_j \}_{j \in J})$ be a D1726: Filtered probability space.
A D5697: Real random process $X : J \to \text{Random}(\Omega \to \mathbb{R})$ is a real martingale on $P$ if and only if

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