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Baire measurability in C (2

)

Grzegorz Plebanek (Uniwersytet Wroc̷lawski)

joint work with A. Avil´es and J. Rodr´ıguez (Universidad de Murcia)

Integration, Vector Measures and Related Topics IV March 2011

Terminology and notation

For any family 𝒜 ⊆ 𝒫(X ) we write 𝜎(𝒜) for the 𝜎-algebra generated by 𝒜.

If ℱ ⊆ ℝ^X is a family of functions ten 𝜎(ℱ ) denotes the 𝜎-algebra generated by ℱ , i.e. the least 𝜎-algebra making all f ∈ ℱ measurable.

Baire and Borel sets

In every completely regular topological space X there are two natural 𝜎-algebras:

Bor (X ) generated by all open sets, and

Baire(X ) generated by all continuous functions X → ℝ.

Baire(X ) = Bor (X ) whenever X is a metric space, in general

Banach spaces C (K )

For a compact space K we can equipp C (K ) with three natural topologies

(C (K ), ∣∣ ⋅ ∣∣);

(C (K ), weak);

(C (K ), 𝜏p).

We can discuss five 𝜎-algebras on C (K ). Recall that Baire(C (K ), 𝜏_p) = 𝜎(𝛿_x : x ∈ K ), where 𝛿_x(g ) = g (x ) Baire(C (K ), weak) = 𝜎(𝜇 : 𝜇 ∈ C (K )^∗), where 𝜇(g ) =∫ g d𝜇.

Borel structures in C (2^𝜅)

Bor (C (2^𝜅), 𝜏_p) = Bor (C (2^𝜅), weak) = Bor (C (2^𝜅)),

for every 𝜅 because C (2^𝜅) has a 𝜏p-Kadec renorming (Edgar).

Baire structures in C (2^𝜅) for 𝜅 ≤ 𝔠 Baire(C (2^𝜅), 𝜏_p) = Baire(C (2^𝜅), weak),

for 𝜅 ≤ 𝔠 because every probability measure 𝜇 on 2^𝔠 is a weak^∗-limit 𝜇 = lim_n(1/n)∑

i ≤n𝛿_xi for some sequence x_i ∈ 2^𝜅 (Fremlin).

For 𝜅 ≤ 𝔠 we have thus the Baire 𝜎-algebra on C (2^𝜅) and its Borel 𝜎-algebra.

Theorem

Baire(C (2^𝜔1), 𝜏 ) = Bor (C (2^𝜔1), 𝜏 ) and, consequently, all the five

Why Baire(C (2^𝜔1), 𝜏_p) = Bor (C (2^𝜔1), 𝜏_p)?

Lemma

Suppose that K is such a compact space that for every n ∈ ℕ and every closed F ⊆ Kⁿ, F is a decreasing intersection of a sequence (Fp)_p∈ℕ of closed separable subspaces Fp ⊆ Kⁿ.

Then Baire(C (K ), 𝜏_p) = Bor (C (K ), 𝜏_p).

Lemma

Every closed F ⊆ 2^𝜔1 is a decreasing intersection of a sequence (Fp)_p∈ℕ of closed separable subspaces Fp ⊆ 2^𝜔1.

Kunen cardinals

𝜅 is a Kunen cardinal if 𝒫(𝜅) ⊗ 𝒫(𝜅) = 𝒫(𝜅 × 𝜅), i.e.

𝜎({A × B : A, B ⊆ 𝜅}) contains all subsets of 𝜅 × 𝜅.

If 𝜅 is Kunen then 𝜅 ≤ 𝔠.

𝜔₁ is a Kunen cardinal.

𝔠 is Kunen cardinal under MA + non CH, but, consistently, 𝔠= 𝜔₂ is not Kunen.

If 𝜅 is a Kunen cardinal then there is no universal measure on 𝒫(𝜅).

Fremlin’s result and a corollary

Baire(l₁(𝜅), weak) = Bor (l₁(𝜅), weak) iff 𝜅 is a Kunen cardinal.

If Baire(C (2^𝜅), 𝜏p) = Bor (C (2^𝜅), 𝜏p) then 𝜅 is a Kunen cardinal.

Theorem - the main result

Baire(C (2^𝜅), 𝜏p) = Bor (C (2^𝜅), 𝜏p) iff 𝜅 is a Kunen cardinal.

Corollary

C (2^𝜅) is measure-compact whenever 𝜅 is a Kunen cardinal.

A Banach space E is measure compact if for every weakly measurable f : Ω → E there is a Bochner measurable g : Ω → E such that x^∗g = x^∗f 𝜇-a.e. (for any probability space (Ω, Σ, 𝜇)).

Equivalently, for every finite measure 𝜈 on Baire(E , weak) there is a separable subspace E₀ such that 𝜇^∗(E₀) = 𝜇(E ).

Remark

Assuming the absence of weakly inaccessible cardinals, C (2^𝜅) is measure-compact for any 𝜅. (Plebanek [1991])

Corollary

Under MA + non CH, Bor (2^𝜔1) is countable generated.