Topological completeness of spaces of measures

We prove that the functors P-R and P-tau of Radon and tau-additive probability measures, respectively, preserve neither the real-completeness nor the Dieudonne completeness of Tychonoff spaces. We suggest conditions under which Martin's axiom implies that P-tau preserves real-complete spaces, absolute extensors, and Tychonoff bundles. These last results cannot be obtained without additional set-theoretic assumptions.