Spaces of measures on metrizable spaces

Let P(X) be the space of probability measures on a space X and let PS(X), PR(X), PZ(X) and Pn(X) be subspaces of P(X) consisting of measures with separable supports, compact supports, finite supports and with supports consisting of at most n points, respectively. It is shown that, for a quadruple (T, X, Y, Z) of separable metrizable spaces, (P(T), P(X), PR(Y), PZ(Z)) ≈ (Q, s, Σ, σ) if and only if T is compact, X ⊂≠ T is Gδ in T, Y is open in T, Y ∼ N and Z is σ-fd-compact and dense in T, where ≈ means “is homeomorphic to”, Q = [−1, 1]ω is the Hilbert cube, s = (−1, 1)ω the pseudo-interior of Q, Σ = {(xi) ϵ Q ¦ sup ¦xi¦ < 1} the radial-interior of Q and σ = {(xi) ϵ s ¦ xi = 0 except for finitely many i}. In case X is nonseparable, we prove that PS(X) is homeomorphic to a Hilbert space ℓ2(A) if and only if X is completely metrizable and dens X = card A (⪢ ℵ0). Moreover it is proved that (PS(ℓ2(A)),PZ(ℓ2(A))) ≈ (ℓ2(A)ω, ℓ2(A)ωf) and Pn(ℓ2(A)) ≈ ℓ2(A) for an arbitrary infinite set A and that PZ(E) ≈ Pn(E) ≈ E for any pre-Hilbert space E which is homeomorphic to Eωf = {(xi) ϵ Eω ¦ xi = 0 except for finitely many i ϵ N} ⊂ Eω. e similar results for the space M+(X) of nonnegative measures of X.