On measures on convex Corson compacta

Grzegorz Plebanek (University of Wroclaw)

Given κ, the Corson space Σ(R^{κ}) is

Σ(R^{κ}) = {x ∈ R^{κ} : |{α < κ : x(α) ≠ 0}| ≤ ω}.

A Corson compactum is any space which is homeomorphic to a compact subset of Σ(R^{κ}) for some κ.

Remark. A Corson compact K is separable if and only if K is metrizable.

P (K) denotes the set of all probability Radon measures on K. Every μ ∈ P (K) is a Borel measure on K, such that for any Borel set B

μ(B) = sup {μ(F) : F ⊆ B, F compact }.

P (K) is equipped with the weak^{∗} topology inherited from C(K)^{∗}. Basic neighbourhoods of μ ∈ P (K) are of the form

{ν ∈ P (K) : | ∫ g_{i} dμ−∫ g_{i} dν| < ε for i ≤ k}.

A measure μ is called separable or of countable type if L_{1}(μ) is a separable Banach space.

Theorem. If MA(ω_{1}) holds then every Radon measure on a Corson compact space has a metrizable support.

Fact. Let K be a compact convex subset of Σ(R^{κ}) . For α < κ, π_{α} is the usual projection. Every μ ∈ P (K) has a barycentre b_{μ} ∈ K, i.e.

∫ π_{α} dμ = π_{α}(b_{μ}) = b_{μ}(α).

Theorem Let K be a compact convex subset of Σ(R^{κ}) .

(1) If μ ∈ P (K) is of countable type then μ has a metrizable support.

(2) If K is included in Σ(R_{+}^{κ} ) then every μ ∈ P (K) has a metrizable support.

Problem. (O. Kalenda) Can we prove in ZFC that every μ ∈ P (K) has a metrizable suupport whenever K is compact and convex subset of Σ(R^{κ}) ?

Theorem 1. [CH] There are a nonseparable 0–dimensional Corson compact space K, a strictly positive measure μ ∈ P (K) of uncountable type, and a family (g_{α})_{α<ω_{1}} ⊆ C(K) of norm one functions such that for every ν ∈ P (K), the set

{ξ < ω_{1} : ∫ g_{ξ} dν ≠ 0},

is countable.

Theorem 2. A compact space K has a strictly positive Radon measure if and only if the space (P (K), weak^{∗}) has a strictly positive Radon measure.

Theorem 3. [CH] There exists a compact convex nonseparable subspace L of Σ(R^{ω_{1}}) , and a strictly positive measure λ ∈ P (L) (which has no metrizable support).

Φ(ν) = (∫ g_{ξ} dν)_{ξ<ω_{1}}
Φ : P (K) → Σ(R^{ω_{1}})

Φ is an affine continuous mapping, and hence L = Φ(P (K)) is a compact convex subset of Σ(R^{ω_{1}}) .

P (K) has a strictly positive Radon measure Υ by Theorem 2. The measure λ = Φ(Υ) is strictly positive on L.

Claim. There is an algebra A ⊆ Ba{0, 1}^{ω_{1}} such that

1. A = ⋃_{α<ω_{1}} A_{α}, |A_{α}| = ω, A_{β} ⊆ A_{α} for β < α < ω_{1}, A_{0} is some nonatomic algebra;

2. λ(A) > 0 for ∅ ≠ A ∈ A;

3. A_{α} = alg(⋃_{β<α} A_{β} ∪ {H_{α}^{0}, H_{α}^{1}}), H_{α}^{0} ∩ H_{α}^{1} = ∅, F_{α} := H_{α}^{0} ∪ H_{α}^{1} ≠ ∅, and for every β < α and C ∈ A_{β} λ(C ∩ H_{α}^{0}) = λ(C ∩ H_{α}^{1});

4. if (A_{n}) ⊆ A is decreasing and lim λ(A_{n}) = 0 then there is α < ω_{1} such that for all η > α we have A_{n} ∩ F_{η} = ∅ for large n.

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