We define $\Delta$-Baire spaces. If a paratopological group $G$ is space, then topological group. Locally pseudocompact spaces, Baire $p$-spaces, $\Sigma$-spaces, products of \v{C}ech-complete spaces are

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...

For any countable group satisfying the \weak Rohlin property", and for each dynamical property, the set of-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z d ; indeed, all countable discrete amenable groups. For an arbitrary countable group, let A be the set of-actions on the unit circle Y. We establish an Equival...

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...

A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...

We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures. Introduction. The Borel reducibility theory of Pol...

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 kn...

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall’s condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theore...