Borel Extensions of Baire Measures in Zfc
نویسنده
چکیده
We prove: (1) Every Baire measure on the Kojman-Shelah Dowker space [10] admits a Borel extension. (2) If the continuum is not a real-valued measurable cardinal then every Baire measure on the M. E. Rudin Dowker space [16] admits a Borel extension. Consequently, Balogh’s space [3] remains as the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.
منابع مشابه
Regularity of group valued Baire and Borel measures
It is known that a real valued measure (1) on the a-ring of Baire sets of a locally compact Hausdorff space, or (2) on the Borel sets of a complete separable metric space is regular. Recently Dinculeanu and Kluvanek used regularity of non-negative Baire measures to prove that any Baire measure with values in a locally convex Hausdorff topological vector space (TVS) is regular. Subsequently a di...
متن کاملContinuous Images of Closed Sets in Generalized Baire Spaces
Let κ be an uncountable cardinal with κ = κ<κ. Given a cardinal μ, we equip the set κμ consisting of all functions from κ to μ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than κ. We prove results that allow us to separate several classes of subsets of κκ that consist of continuous images of closed subsets of spaces of the form κμ. I...
متن کاملUniversal Functions
A function of two variables F (x, y) is universal if for every function G(x, y) there exists functions h(x) and k(y) such that G(x, y) = F (h(x), k(y)) for all x, y. Sierpiński showed that assuming the Continuum Hypothesis there exists a Borel function F (x, y) which is universal. Assuming Martin’s Axiom there is a universal function of Baire class 2. A universal function cannot be of Baire cla...
متن کاملWadge Hierarchy of differences of Co-analytic Sets
We begin the fine analysis of non Borel pointclasses. Working in ZFC + ̃ 11), we describe the Wadge hierarchy of the class of increasing differences of coanalytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets. Introduction. Collections of substets of the Baire space, the "logician’s reals", that are closed under continuous preimages have always been...
متن کاملExtending Baire property by uncountably many sets
We prove that if ZFC is consistent so is ZFC + “for any sequence (An) of subsets of a Polish space 〈X, τ〉 there exists a separable metrizable topology τ ′ on X with B(X, τ) ⊆ B(X, τ ′), MGR(X, τ ′) ∩ B(X, τ) = MGR(X, τ) ∩B(X, τ) and An Borel in τ ′ for all n.” This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets...
متن کامل