Martingale Convergence of Generalized Conditional Expectations

We prove that the martingale convergence theorem for generalized conditional expectations in von Neumann algebras holds in the weak topology without restrictions. The situation is therefore different fom the strong topology case, where there are restrictive conditions which distinguish between increasing and decreasing sequences of von Neumann algebras. Moreover, known couterexamples show that in the decreasing case the strong martingale convergence theorem might not hold.