The following results are announced without proofs. For a martingale the uniform integrability, the existence of a limit in \(L^ 1\) and the fact that it consists of conditional mean values are equivalent. Its \(L^ 2\)- boundedness, its \(L^ 2\)-convergence and the fact that it consists of conditional mean values of an element in \(L^ 2\) are also equivalent. A corresponding \(L^{\infty}\)-version is also considered. Every \(L^ 1\)- bounded supermartingale converges a.e. Every \(L^ 2\)-bounded martingale converges s-a.e. to its limit in \(L^ 2\). All martingales are with discrete time.
The s-a.e. convergence implies the a.e. one. The conditional mean value, the a.e. and the s-a.e. convergence are defined in the paper, but all these concepts are relative to a JBW algebra and to a normal faithful finite trace on it, as defined in the author’s paper, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk 1981, No.5, 3-6 (1981; Zbl 0483.46048) and in T. A. Sarymsakov and the author, Dokl. Akad. Nauk SSSR 249, 789-792 (1979; Zbl 0442.46037). The recent author’s book, Classification and representations of ordered Jordan algebras. Tashkent (1986), ch. 4, is also useful.