Convex Corson compacta and Radon measures

Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of Sigma(R-omega1), and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no G(delta)-points.