Extending Baire measures to Borel measures

Transcription

1 Extending Baire measures to Borel measures Menachem Kojman Ben-Gurion University of the Negev MCC, Bedlewo 2007 p. 1/1

2 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? MCC, Bedlewo 2007 p. 2/1

3 Mařik s theorem Theorem 1 (Mařik 1957) If a normal space X is countably paracompact then every Baire measure µ extends to a unique, inner regular, Borel measure µ. [ CPC X = D n Dn = U n D n s.t. ] U n = 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.) MCC, Bedlewo 2007 p. 3/1

4 Mařik and quasi-mařik spaces A normal X is Mařik if every Baire measure extends to an i.r. Borel measure. A normal X is quasi-mařik if every Baire measure extends to some Borel measure. The extension problem: are there non quasi-mařik (Dowker) spaces? Ohta and Tamano 1990: are there quasi-mařik Dowker spaces? MCC, Bedlewo 2007 p. 4/1

5 Consistent answers Fremlin (Budapest Zoo café 1999): The axiom implies the existence of a de-caux type Dowker space on ℵ 1 which is not quasi-mařik. Aldaz 1997: The axiom implies the existence of a de-caux type Dowker space on ℵ 1 which is quasi-mařik, non-mařik. Fremlin: is there a ZFC example of a non-quasi-mařik space? MCC, Bedlewo 2007 p. 5/1

6 ZFC Dowker spaces A ZFC Dowker space X R was constructed in ZFC by M. E. Rudin in Its cardinality is (ℵ ω ) ℵ 0. For over 20 years this was the only Dowker space in ZFC. P. Simon 1971: X R is not Mařik. Balogh 1996: X B of cardinality 2 ℵ 0. Constructed by transfinite induction of length 2 2ℵ 0. Kojman-Shelah 1998: A closed subspace X KS X R of cardinality ℵ ω+1. Constructed with a PCF-theory scale. MCC, Bedlewo 2007 p. 6/1

7 The results In a joint work with H. Michalewski, to appear on Fundamenta: X KS is quasi-mařik. This gives a ZFC answer to Ohta and Tamano. In particular, it is not a ZFC counter-example to the measure extension problem. X R is also not a ZFC counter example because if the continuum is not real valued measurable, then X R is quasi-mařik. This leaves X B as the only candidate at the moment to be a ZFC counter-example. MCC, Bedlewo 2007 p. 7/1

8 The set theoretic aspect It is not known whether a Dowker space on ℵ 1 has to exist or not; but it is known that one exists on ℵ ω+1. What is the difference between these cardinals? 2 ℵ 0 (ℵ ω ) ℵ 0 ω ω n ω n b = b(ω ω,< ) ω 1 b( n ω n,< ) = ℵ ω+1 d = d(ω ω,< ) unbounded d( n ω n,< ) < ℵ ω4 (ℵ ω ) ℵ 0 = 2 ℵ 0 d( n ω n,< MCC, Bedlewo 2007 p. 8/1

9 Naming the parts of X R P = n ω n T = { f P : ( n) cff(n) > ℵ 0 } X R = { f T : ( m)( m) cff(n) ℵ m } The topology on X R is the box product topology. A basic clopen set is of the form (f,g] where f < g are in P. X R is clearly a p-space, that is, every G δ set in X R is open. Hence, Baire = clopen. MCC, Bedlewo 2007 p. 9/1

10 Rudin Spaces Suppose g T \ X R. X g = X R (0,g] A Rudin space is a closed X X g for some g T \ X R, which is also cofinal in (X g, ). Rudin spaces are closed in X R hence are normal. Suppose g T \ X R. f X g is m-normal in X g if cfg(n) ℵ m f(n) = g(n) and cfg(n) > ℵ n cff(n) = ℵ m. MCC, Bedlewo 2007 p. 10/1

11 m-clubs An m-club is a set of m-normal elements which is cofinal and closed under suprema of length ℵ m. A closed D X g is cofinal iff it contains an m club for all m m 0 for some m 0. 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 X g clopen, then D contains an end segment of X g. 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. MCC, Bedlewo 2007 p. 11/1

12 Cofinal Baire measures Suppose X X g is closed and cofinal in X g. A cofinal Baire measure is a Baire measure µ which satisfies µ(y ) = r iff Y is cofinal in X g 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 X g, for m m 0. MCC, Bedlewo 2007 p. 12/1

13 General Baire measures Theorem 2 For every Baire measure µ on a Rudin space X, if min{ X, 2 ℵ 0 } is not real valued measurable, there are countably many pairwise disjoint Rudin subspaces X n X and a countable Y = {f m : m ω} so that µ X n is a cofinal Baire measure on X n and µ = n µ n + m µ {f m }. Corollary 1 X KS is quasi-mařik. X R is quasi Mařik unless the continuum is real-valued measurable; so it is not a ZFC counter-example to the extension problem. MCC, Bedlewo 2007 p. 13/1

14 Concluding remarks and problems The small Dowker space problem. Is it consistent that no counter example to the measure extension problem exists below ℵ ω? In ℵ 1? MCC, Bedlewo 2007 p. 14/1