Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) <= Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.