Martingale submartingale and supermartingale

Assume AC The Axiom of Choice. An integrable adapted real process $X$ Adapted and integrable stochastic process is a martingale, submartingale, or supermartingale when, respectively, for every $n\ge 0$, $E[X_{n+1}\mid {\mathcal{F}}_n]=X_n,E[X_{n+1}\mid {\mathcal{F}}_n]\ge X_n,E[X_{n+1}\mid {\mathcal{F}}_n]\le X_n\text{almost surely}\mathrm{.}$ These are three separate conditions, with the convention that a submartingale has conditional future mean at least its present value. The conditional expectations are the almost-sure classes of Conditional expectation as an ae class; that supplier inherits AC from its Radon–Nikodym existence proof. The definition itself selects no representative and assumes no completed filtration.

The condition is imposed at each adjacent pair of times. An all-pairs formulation requires a proof. No boundedness or independence is part of this definition, and no trivial or deterministic initial value is imposed.