Pseudo-martingales

For a probability space (Ω, ℱ, P)and a filtration (Un) in Ω,we consider the sequences (Xn) of random variables satisfying the condition E (Xn+1 − Xn | Un) = 0, n = 1, 2, ... In general, the proces (Xn) is not required to be (Un) adapted and it is called a pseudo-martingale. We indicate simple and natural conditions implying a good asymptotic behaviour of pseudo-martingales. For example: let (Xn, Un) be a uniformly integrable pseudo-martingale with Un ↗ ℱ. Then Xn → X weakly in L1 where [formula]. Some approximation results for σ-fields are obtained with implications to pseudo-martingales. A number of examples is collected.