The author considers strong operator and weak operator convergence of regular generalized martingales in a von Neumann algebra and their norm and weak convergence in noncommutative \(L^p\)-spaces over the algebra, with \(1\leq p<\infty\). The starting point is a von Neumann algebra \(M\) with a monotone net of subalgebras \(M_\alpha\), and an f.n.s. weight \(\varphi\) on \(M\) which is still semifinite when restricted to the \(M_\alpha\)’s. One deals with pointwise convergence of generalized conditional expectations from \(M\) into \(M_\alpha\) or their canonical extensions to maps from \(L^p(M)\) into \(L^p(M_\alpha)\), composed with the natural embeddings of \(M_\alpha\) into \(M\) or their extensions to maps from \(L^p(M_\alpha)\) into \(L^p(M)\). To avoid unnecessary complications, only the symmetric embedding is used.
Besides gathering most old results on the subject, the author completes the picture, especially for the case of non-finite measures. A surprising fact is that, in the increasing case, the norm limit of a regular martingale \((E_\alpha(x))\) in an \(L^p\)-space may be different from \(E_\infty(x)\), where \(E_\infty\) corresponds to the generalized conditional expectation on the limiting subalgebra \(M_\infty\). Another result shows that weak convergence of martingales in \(L^p\)-spaces implies their norm convergence. The differences between the increasing and decreasing cases are clearly indicated.