3,058 research outputs found
Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations
There is a fascinating interplay and overlap between recursion theory and
descriptive set theory. A particularly beautiful source of such interaction has
been Martin's conjecture on Turing invariant functions. This longstanding open
problem in recursion theory has connected to many problems in descriptive set
theory, particularly in the theory of countable Borel equivalence relations.
In this paper, we shall give an overview of some work that has been done on
Martin's conjecture, and applications that it has had in descriptive set
theory. We will present a long unpublished result of Slaman and Steel that
arithmetic equivalence is a universal countable Borel equivalence relation.
This theorem has interesting corollaries for the theory of universal countable
Borel equivalence relations in general. We end with some open problems, and
directions for future research.Comment: Corrected typo
Integrable Functions Versus a Generalization of Lebesgue Points in Locally Compact Groups
The author is thankful to the referee for his valuable
comments and suggestions that led to an improvement of the paper.
He also owes to Prof. M. N. Mukherjee of the Deptt. of Pure Mathematics,
Calcutta University, for the present linguistically improved version.Here in this paper we intend to deal with two questions: How large is a “Lebesgue Class” in the topology of Lebesgue integrable functions, and also what can be said regarding the topological size of a “Lebesgue set” in R?, where by a Lebesgue class (corresponding to some x in R) is meant the collection of all Lebesgue integrable functions for each of which the point x acts as a common Lebesgue point, and, by a Lebesgue set (corresponding to some Lebesgue integrable function f ) we mean the collection of all ebesgue points of f.
However, we answer these two questions in a more general setting where in place of Lebesgue integration we use abstract integration in locally compact Hausdorff topological groups
Pincherle's theorem in Reverse Mathematics and computability theory
We study the logical and computational properties of basic theorems of
uncountable mathematics, in particular Pincherle's theorem, published in 1882.
This theorem states that a locally bounded function is bounded on certain
domains, i.e. one of the first 'local-to-global' principles. It is well-known
that such principles in analysis are intimately connected to (open-cover)
compactness, but we nonetheless exhibit fundamental differences between
compactness and Pincherle's theorem. For instance, the main question of Reverse
Mathematics, namely which set existence axioms are necessary to prove
Pincherle's theorem, does not have an unique or unambiguous answer, in contrast
to compactness. We establish similar differences for the computational
properties of compactness and Pincherle's theorem. We establish the same
differences for other local-to-global principles, even going back to
Weierstrass. We also greatly sharpen the known computational power of
compactness, for the most shared with Pincherle's theorem however. Finally,
countable choice plays an important role in the previous, we therefore study
this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi
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