Best approximations in metric spaces

Let E be a Kothe space of real random variables (r.v.) on ([0, 1], B, lambda), rearrangement invariant and weakly sequentially complete. Let M be a metric separable space whose closed and bounded parts are compact. We show that, for every sub-algebra F and every M-values r.v. X is an element of E(M), i.e. For All x is an element of M, d(X,x) is an element of E, there exists a best approximant of X for F, that is to say: a r.v. Y is an element of E(M), F-measurable such that parallel to d(X, Y)parallel to(E) less than or equal to parallel to d(X, Z)parallel to(E), For All Z, F-measurable. The key consists in proving that E(M) is also, in a certain way, weakly sequentially complete. We give some consequences. (C) Academie des Sciences/Elsevier, Paris.