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
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
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
minimal size of a local base at x.
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
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.
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
Proof. Assume 𝜒(𝜇, M) ≥ 𝜅 for every 𝜇 ∈ M. Construct
g𝜉 ∈ C (K ) and 𝜇𝜉 ∈ M for 𝜉 ≤ 𝜅 so that
(i) for 𝛼 < 𝜉 ≤ 𝜅 and f ∈ ℱ𝛼 = {g𝛽 , ∣g𝛽 − g𝛽 ′ ∣ : 𝛽, 𝛽 ′ ≤ 𝛼} we
have 𝜇𝜉 (f ) = 𝜇𝛼 (f );
(ii) for 𝜉 < 𝜅 we have inf 𝛼<𝜉 𝜇𝜉 (∣g𝜉 − g𝛼 ∣) > 0.
Then the family {g𝜉 : 𝜉 < 𝜅} will witness that the measure 𝜇𝜅 is of
type 𝜅. At the step 𝜉 we consider
M𝜉 =
∩
{𝜇 ∈ M : 𝜇(f ) = 𝜇𝛼 (f ) for f ∈ ℱ𝛼 } =
∕ ∅.
𝛼<𝜉
M𝜉 has more than one ∪
element. Take different 𝜇, 𝜇′ ∈ M𝜉 and set
′
𝜇𝜉 = (𝜇 + 𝜇 )/2. Then 𝛼<𝜉 ℱ𝛼 is not dense in L1 (𝜇𝜉 ), and we
can find g𝜉 as in (ii).
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
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 G𝛿 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.
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
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 .
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
Cardinal 𝔭
𝔭 is defined so that∩whenever 𝛾 < 𝔭 and (N𝜉 )𝜉<𝛾 is a family of
subsets of ℕ with 𝜉∈I N𝜉 infinite for every finite I ⊆ 𝛾 then there
is an infinite N ⊆ ℕ such that N ⊆∗ N𝜉 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 = limn an for some an ∈ A.
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures
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 (𝜇n )n in P(K ) contains a weak ∗ converning
block subsequence, or
there is 𝜇 ∈ P(K ) of type ≥ 𝔭.
Proof. For a fixed (𝜇n )n in P(K ) consider
∩
M=
conv{𝜇k : k ≥ n}.
n
If 𝜇 ∈ M and 𝜒(𝜇, M) < 𝔭 then there is a block converging
subsequence. Otherwise we apply our dichotomy for 𝜅 = 𝔭.
Grzegorz Plebanek (Uniwersytet Wroc̷lawski)
A dichotomy for the spaces of probability measures