The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov’s 0-1 law that for any property which may or may not be satisfied by any given Turing degree, the satisfying class will either be of Lebesgue measure 0 or 1, so long as it is measurable. So either the typical degree satisfies the prop...

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 show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is א0. We also show that if add(null) = c, then the universally measurable chromatic number of every treei...

There is a natural topology on the symmetric group on an infinite set Ω . If Ω is countable , the topology is derived from a metric , and the group is complete . This paper gives an account of this topology , including translations of topological concepts for permutation groups , the use of Baire category and Haar measure , and some results about confinitary permutation groups which are motivat...

Let A be a commutative unital C-algebra and let  denote its Gelfand spectrum. We find some necessary and sufficient conditions for a nondegenerate representation of A to be unitarily equivalent to a multiplicative representation on a space L(Â, μ), where μ is a positive measure on the Baire sets of Â. We also compare these conditions with the multiplicity-free property of a representation.

Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resource-bounded extension: if a class X has ∆-strong dimension less than 1, then X is ∆-meager. This has the applications of explaining some of Lutz’s simultaneous ∆-meager, ∆-measure 0 results and providing a new proof of a Gu’s strong dimension result on infinitely...

We study Baire category for downward-closed subsets of 2ω , showing that it behaves better in this context than for general subsets of 2ω . We show that, in the downward-closed context, the ideal of meager sets is prime and b-complete, while the complementary filter is g-complete. We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results fo...

For a multidimensional control problem (P)K involving controls u ∈ L∞, we construct a dual problem (D)K in which the variables ν to be paired with u are taken from the measure space rca (Ω, B) instead of (L∞)∗. For this purpose, we add to (P)K a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

In this paper, the concepts of pairwise fuzzy Baire bitopological spaces are introduced and characterizations of pairwise fuzzy Baire bitopological spaces are studied. Several examples are given to illustrate the concepts introduced in this paper.

We study the functions whose composition with Baire class one functions are Baire class one functions. We first prove some characterizations of such functions, then investigate a subclass of such functions which are defined in a natural way.