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 MCC, Bedlewo 2007 – p. 1/1

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

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 ∗ . � � � � CPC X = ∀ D n D n = ∅ ⇒ ∃ 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

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? MCC, Bedlewo 2007 – p. 4/1

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-Maˇ rik. Aldaz 1997: The axiom ♣ implies the existence of a de-Caux type Dowker space on ℵ 1 which is quasi-Maˇ rik, non-Maˇ rik. Fremlin: is there a ZFC example of a non-quasi-Maˇ rik space? MCC, Bedlewo 2007 – p. 5/1

ZFC Dowker spaces A ZFC Dowker space X R was constructed in ZFC by M. E. Rudin in 1970. 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 Maˇ rik. Balogh 1996: X B of cardinality 2 ℵ 0 . Constructed by transfinite induction of length 2 2 ℵ 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

The results In a joint work with H. Michalewski, to appear on Fundamenta: X KS 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. X R is also not a ZFC counter example because if the continuum is not real valued measurable, then X R is quasi-Maˇ rik. This leaves X B as the only candidate at the moment to be a ZFC counter-example. MCC, Bedlewo 2007 – p. 7/1

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 n ω n , < ∗ ) = ℵ ω +1 b ( � d = d ( ω ω , < ∗ ) unbounded n ω n , < ∗ ) < ℵ ω 4 d ( � ( ℵ ω ) ℵ 0 = 2 ℵ 0 × d ( � n ω n , < MCC, Bedlewo 2007 – p. 8/1

Naming the parts of X R � P = ω n +2 + 1 n � � f ∈ P : ( ∀ n ) cf f ( n ) > ℵ 0 T = X R = � � f ∈ T : ( ∃ m )( ∀ m ) cf f ( 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

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 cf g ( n ) ≤ ℵ m ⇒ f ( n ) = g ( n ) and cf g ( n ) > ℵ n ⇒ cf f ( n ) = ℵ m . MCC, Bedlewo 2007 – p. 10/1

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

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

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 } . X KS is quasi-maˇ Corollary 1 rik. X R is quasi Maˇ rik 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

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