Extending Baire measures to Borel measures Menachem Kojman - - PowerPoint PPT Presentation

Extending Baire measures to Borel measures

Menachem Kojman Ben-Gurion University of the Negev

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Beginning

A Baire measure is a probability measure on the Baire sigma-algebra over a normal Hausdorff space X. A Borel measure is a probability measure on the Borel sigma-algebra over a normal Hausdorff space X. The extension problem: given a Baire measure µ, is there a Borel measure µ∗ so that µ ⊆ µ∗?

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Maˇ rik’s theorem

Theorem 1 (Maˇ rik 1957) If a normal space X is countably paracompact then every Baire measure µ extends to a unique, inner regular, Borel measure µ∗. CPCX = ∀Dn Dn = ∅ ⇒ ∃Un ⊇ Dn s.t.

• Un = ∅
• Thus, if some Baire measure µ on some normal X does

not extend to a unique, i.r. Borel µ∗, then X is a Dowker

• space. (Dowker spaces originated from Borsuk’s work in

homotopy theory; an equivalent definition of such a space is that its product with [0, 1] is not normal.)

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Maˇ rik and quasi-Maˇ rik spaces

A normal X is Maˇ rik if every Baire measure extends to an i.r. Borel measure. A normal X is quasi-Maˇ rik if every Baire measure extends to some Borel measure. The extension problem: are there non quasi-Maˇ rik (Dowker) spaces? Ohta and Tamano 1990: are there quasi-Maˇ rik Dowker spaces?

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Consistent answers

Fremlin (Budapest Zoo café 1999): The axiom ♣ implies the existence of a de-Caux type Dowker space

• n ℵ1 which is not quasi-Maˇ

rik. Aldaz 1997: The axiom ♣ implies the existence of a de-Caux type Dowker space on ℵ1 which is quasi-Maˇ rik, non-Maˇ rik. Fremlin: is there a ZFC example of a non-quasi-Maˇ rik space?

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ZFC Dowker spaces

A ZFC Dowker space XR was constructed in ZFC by

• M. E. Rudin in 1970. Its cardinality is (ℵω)ℵ0. For over

20 years this was the only Dowker space in ZFC. P . Simon 1971: XR is not Maˇ rik. Balogh 1996: XB of cardinality 2ℵ0. Constructed by transfinite induction of length 22ℵ0. Kojman-Shelah 1998: A closed subspace XKS ⊆ XR

• f cardinality ℵω+1. Constructed with a PCF-theory

scale.

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The results

In a joint work with H. Michalewski, to appear on Fundamenta: XKS is quasi-Maˇ

• rik. This gives a ZFC answer to Ohta

and Tamano. In particular, it is not a ZFC counter-example to the measure extension problem. XR is also not a ZFC counter example because if the continuum is not real valued measurable, then XR is quasi-Maˇ rik. This leaves XB as the only candidate at the moment to be a ZFC counter-example.

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The set theoretic aspect

It is not known whether a Dowker space on ℵ1 has to exist

• r not; but it is known that one exists on ℵω+1. What is the

difference between these cardinals? 2ℵ0 (ℵω)ℵ0 ωω

• n ωn

b = b(ωω, <∗)=ω1 b(

n ωn, <∗) = ℵω+1

d = d(ωω, <∗) unbounded d(

n ωn, <∗) < ℵω4

(ℵω)ℵ0 = 2ℵ0 × d(

n ωn, <

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Naming the parts of XR

P =

• n

ωn+2 + 1 T =

• f ∈ P : (∀n) cff(n) > ℵ0
• XR =
• f ∈ T : (∃m)(∀m) cff(n) ≤ ℵm
• The topology on XR is the box product topology. A

basic clopen set is of the form (f, g] where f < g are in P. XR is clearly a p-space, that is, every Gδ set in XR is

• pen. Hence, Baire = clopen.

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Rudin Spaces

Suppose g ∈ T \ XR. Xg = XR ∩ (0, g] A Rudin space is a closed X ⊆ Xg for some g ∈ T \ XR, which is also cofinal in (Xg, ≤). Rudin spaces are closed in XR hence are normal. Suppose g ∈ T \ XR. f ∈ Xg is m-normal in Xg if cfg(n) ≤ ℵm ⇒ f(n) = g(n) and cfg(n) > ℵn ⇒ cff(n) = ℵm.

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m-clubs

An m-club is a set of m-normal elements which is cofinal and closed under suprema of length ℵm. A closed D ⊆ Xg is cofinal iff it contains an m club for all m ≥ m0 for some m0. Fodor lemma for m-clubs: if f > F(f) ∈ P for all f in some m-club D, then there is a fixed h ∈ P so that {f ∈ D : F(f) < h} is m-stationary. If D ⊆ Xg clopen, then D contains an end segment of Xg. Cofinal clopen sets form a σ-ultrafilter of clopen sets. Closed cofinal sets are just a filter of closed sets. All Rudin spaces are Dowker.

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Cofinal Baire measures

Suppose X ⊆ Xg is closed and cofinal in Xg. A cofinal Baire measure is a Baire measure µ which satisfies µ(Y ) = r iff Y is cofinal in Xg for some constant r ∈ (0, 1]. Cofinal Baire measures extend to Borel measures, but not to inner regular Borel measures. The extension: µ∗(B) = r iff B contains an m-club of Xg, for m ≥ m0.

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General Baire measures

Theorem 2 For every Baire measure µ on a Rudin space X, if min{|X|, 2ℵ0} is not real valued measurable, there are countably many pairwise disjoint Rudin subspaces Xn ⊆ X and a countable Y = {fm : m ∈ ω} so that µ ↾ Xn is a cofinal Baire measure on Xn and µ =

n µn + m µ{fm}.

Corollary 1 XKS is quasi-maˇ rik. XR is quasi Maˇ rik unless the continuum is real-valued measurable; so it is not a ZFC counter-example to the extension problem.

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Concluding remarks and problems

The small Dowker space problem. Is it consistent that no counter example to the measure extension problem exists below ℵω? In ℵ1?

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