Non-archimedean function spaces and the Lebesgue dominated convergence theorem

Let M(X, K) be the non-archimedean Banach space of all additive and bounded K-valued measures on the ring of all clopen subsets of a zero-dimensional compact space X, where K is a non-archimedean non-trivially valued complete field. It is known that M( X, K) is isometrically isomorphic to the dual of the Banach space C(X,K) of all continuous K-valued maps on X with the sup-norm topology. Does the non-archimedean Lebesgue Dominated Convergence Theorem hold for the space M (X, K)? Only in the trivial case! We show (Theorem 2) that for every sequence (f(n))(n) C(X, K) such that f(n)(x) -> 0 for all x is an element of X and vertical bar vertical bar f(n)vertical bar vertical bar <= 1 for all n is an element of N, one has integral x f(n)d mu -> 0 for each mu is an element of M(X, K) iff Xis finite. In the second part we characterize (Theorem 3) weakly Lindelof non-archimedean Banach spaces E with a base as well as Corson sigma(E', E)-compact unit balls in their duals E' (Theorem 17). We also look at the Kunen space from the non-archimedean point of view.