On abstract conditional expectations

Abstract

(i) If C is a sub-a-field of 2, f,(o) -“f(o) p-a.e. and there is a nonnegative real valued function 40 E L’(Q) s.t. I/f,(o)\/ 1 then lim,,, Ezf,(w) = E”(o). This is just the Lebesgue dominated convergence theorem for abstract conditional expectations (see [S]). (ii) If {zm, z},, , are sub-a-fields of 2, V,“= , C, = Z or fi‘I,“=, 2, = C andfEL$(Q) (1 <PC co) then EZn -+L%(“) E’J: Furthermore if p = 1 then EZnf(o) --+’ Ezf(w) p-a.e. These are known as martingale convergence theorems (see [S] and [14, Theorem 51). Together with the general theory of martingales have an increasingly important impact in Banach space theory (see [5, Chap. VI).