) @E?DJO BH JDA IF=?AI B FH>=>EEJO A=IKHAI

A dichotomy for the spaces of probability measures

Grzegorz Plebanek (Uniwersytet Wroc̷lawski)

joint work with Miko̷laj Krupski (IMPAN Warszawa) Set-theoretic Techniques in Functional Analysis

Castro Urdiales, February 2011

Terminology and notation

K denotes a compact Hausdorff space, P(K ) is the family of regular probability measures on Bor (K ); every 𝜇 ∈ P(K ) satisfies 𝜇(B) = sup{𝜇(F ) : F = F ⊆ B} for B ∈ Bor (K ).

P(K ) ⊆ C (K )^∗ is equipped with the weak^∗ topology, generated by mappings P(K ) ∋ 𝜇 → 𝜇(g ), g ∈ C (K ).

Maharam types

A measure 𝜇 ∈ P(K ) is of type 𝜅 if L₁(𝜇) has density 𝜅;

equivalently, 𝜅 is the least cardinality of a family 𝒜 ⊆ Bor (K ) such that inf{𝜇(A△B) : A ∈ 𝒜} = 0 for every B ∈ Bor (K ).

Local character

If X is any topological space and x ∈ X then 𝜒(x , X ) denotes the

Dichotomy

Let M be a compact and convex subset of P(K ). For any 𝜅 of uncountable cofinality, either

there is 𝜇 ∈ M such that 𝜒(𝜇, M) < 𝜅, or there is 𝜇 ∈ M which is of type ≥ 𝜅.

Proof. Assume 𝜒(𝜇, M) ≥ 𝜅 for every 𝜇 ∈ M. Construct g𝜉∈ C (K ) and 𝜇_𝜉∈ M for 𝜉 ≤ 𝜅 so that

(i) for 𝛼 < 𝜉 ≤ 𝜅 and f ∈ ℱ𝛼= {g_𝛽, ∣g_𝛽− g_𝛽^′∣ : 𝛽, 𝛽^′ ≤ 𝛼} we have 𝜇_𝜉(f ) = 𝜇_𝛼(f );

(ii) for 𝜉 < 𝜅 we have inf_𝛼<𝜉𝜇_𝜉(∣g_𝜉− g_𝛼∣) > 0.

Then the family {g_𝜉: 𝜉 < 𝜅} will witness that the measure 𝜇_𝜅 is of type 𝜅. At the step 𝜉 we consider

M_𝜉 = ∩

𝛼<𝜉

{𝜇 ∈ M : 𝜇(f ) = 𝜇_𝛼(f ) for f ∈ ℱ_𝛼} ∕= ∅.

M_𝜉 has more than one element. Take different 𝜇, 𝜇^′ ∈ M_𝜉 and set 𝜇_𝜉 = (𝜇 + 𝜇^′)/2. Then ∪

𝛼<𝜉ℱ_𝛼 is not dense in L₁(𝜇_𝜉), and we can find g as in (ii).

Uniformly regular measures

𝜒(𝜇, P(K )) = 𝜔 means that 𝜇 is a uniformly regular measure, i.e.

there is a countable family 𝒵 of closed G_𝛿 subsets of K such that 𝜇(U) = sup{𝜇(Z ) : Z ⊆ U, Z ∈ 𝒵}

for every open U ⊆ K (R. Pol).

Corollary for 𝜅 = 𝜔₁ (essentially Borodulin-Nadzieja)

Every compact space [without isolated points] K carries either a uniformly regular [nonatomic] measure or a measure of

uncountable type.

Mappings onto cubes, using Fremlin’s and Talagrand’s results For every compact space K ,

assuming MA(𝜔₁), either P(K ) has points of countable character or K can be mapped onto [0, 1]^𝜔1.

either P(K ) has points of character ≤ 𝜔₁ or P(K ) can be mapped onto [0, 1]^𝜔2.

Cardinal 𝔭

𝔭is defined so that whenever 𝛾 < 𝔭 and (N_𝜉)_𝜉<𝛾 is a family of subsets of ℕ with∩

𝜉∈IN_𝜉 infinite for every finite I ⊆ 𝛾 then there is an infinite N ⊆ ℕ such that N ⊆^∗N_𝜉 for every 𝜉 < 𝛾.

When 𝜒(x , X ) < 𝔭

If x is in the closure of a countable set A ⊆ X and 𝜒(x , X ) < 𝔭 then x = limnan for some an∈ A.

Haydon on Grothendieck spaces

If K is an infinite compact space and C (K ) is Grothendieck then K carries a measure of type ≥ 𝔭.

Haydon-Levy-Odell

For any compact space K either

every sequence (𝜇_n)_n in P(K ) contains a weak^∗ converning block subsequence, or

there is 𝜇 ∈ P(K ) of type ≥ 𝔭.

Proof. For a fixed (𝜇_n)_n in P(K ) consider M =∩

n

conv{𝜇_k : k ≥ n}.