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.