scholarly journals On computability and disintegration

2016 ◽  
Vol 27 (8) ◽  
pp. 1287-1314 ◽  
Author(s):  
NATHANAEL L. ACKERMAN ◽  
CAMERON E. FREER ◽  
DANIEL M. ROY
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Limit Operator ◽  
Covering Property ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

scholarly journals On limit theorems for random fields

2009 ◽  
Vol 50 ◽  
Author(s):  
Rimas Banys
Keyword(s):  
Metric Space ◽  
Random Fields ◽  
Limit Theorems ◽  
Characteristic Property ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

A complete separable metric space of functions defined on the positive quadrant of the plane is constructed. The characteristic property of these functions is that at every point x there exist two lines intersecting at this point such that limits limy→x f (y) exist when y approaches x along any path not intersecting these lines. A criterion of compactness of subsets of this space is obtained.

Martin's axiom and Hausdorif measures

1974 ◽  
Vol 75 (2) ◽  
pp. 193-197 ◽  
Author(s):  
A. J. Ostaszewski
Keyword(s):  
Metric Space ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
The Continuum

AbstractA theorem of Besicovitch, namely that, assuming the continuum hypothesis, there exists in any uncountable complete separable metric space a set of cardinality the continuum all of whose Hausdorif h-measures are zero, is here deduced by appeal to Martin's Axiom. It is also shown that for measures λ of Hausdorff type the union of fewer than 2ℵ0 sets of λ-measure zero is also of λ-measure zero; furthermore, the union of fewer than 2ℵ0 λ-measurable sets is λ-measurable.

scholarly journals On constructing completions

2005 ◽  
Vol 70 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Laura Crosilla ◽  
Hajime Ishihara ◽  
Peter Schuster
Keyword(s):  
Set Theory ◽  
Metric Space ◽  
Ordered Set ◽  
Proper Class ◽  
Original Space ◽  
Full Form ◽  
Dedekind Cuts ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

AbstractThe Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

Reverse Mathematics and Π12 Comprehension

2005 ◽  
Vol 11 (4) ◽  
pp. 526-533 ◽  
Author(s):  
Carl Mummert ◽  
Stephen G. Simpson
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Reverse Mathematics ◽  
Second Order ◽  
Converse Statement ◽  
Second Order Arithmetic ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Metrization Theorem

AbstractWe initiate the reverse mathematics of general topology. We show that a certain metrization theorem is equivalent to Π12 comprehension. An MF space is defined to be a topological space of the form MF(P) with the topology generated by {Np ∣ p ϵ P}. Here P is a poset, MF(P) is the set of maximal filters on P, and Np = {F ϵ MF(P) ∣ p ϵ F }. If the poset P is countable, the space MF(P) is said to be countably based. The class of countably based MF spaces can be defined and discussed within the subsystem ACA0 of second order arithmetic. One can prove within ACA0 that every complete separable metric space is homeomorphic to a countably based MF space which is regular. We show that the converse statement, “every countably based MF space which is regular is homeomorphic to a complete separable metric space,” is equivalent to . The equivalence is proved in the weaker system . This is the first example of a theorem of core mathematics which is provable in second order arithmetic and implies Π12 comprehension.

On the Selection of Compact Subsets of Positive Measure from Analytic Sets of Positive Measure

1974 ◽  
Vol 26 (3) ◽  
pp. 665-677 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Metric Space ◽  
Positive Measure ◽  
Difficult Problem ◽  
List Type ◽  
Finite Dimensional ◽  
Fixed Positive Number ◽  
Analytic Sets ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Selection Of

An important but seemingly difficult problem is to decide whether or not an analytic set A of positive h-measure, for some continuous Hausdorff function h, contains a compact subset C of positive h-measure, in every complete separable metric space Ω.By extending some earlier work of R. O. Davies [1], M. Sion and D. Sjerve [8] proved that (i)the selection of the set C is always possible in a σ-compact metric space Ω. More recently Davies [2] has shown that it is always possible to select C(ii)when h(t) = ts, t ≧ 0, for some fixed positive number s,(iii)when Ω is finite dimensional in the sense of [4],(iv)when A has σ-finite h-measure, and(v)when Ω is an ultra metric space.

REVERSE MATHEMATICS OF MF SPACES

2006 ◽  
Vol 06 (02) ◽  
pp. 203-232 ◽  
Author(s):  
CARL MUMMERT
Keyword(s):  
Metric Space ◽  
Closed Subset ◽  
Metric Spaces ◽  
Reverse Mathematics ◽  
General Topology ◽  
Second Order Arithmetic ◽  
Perfect Set ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF (P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF (P) | p ∈ F}. We define a countably based MF space to be a space of the form MF (P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to [Formula: see text] over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over [Formula: see text] to the proposition that [Formula: see text] is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to [Formula: see text] over [Formula: see text].

Logic and computation, Proceedings of a workshop held at Carnegie Mellon University, June 30–July 2, 1987, edited by Wilfried Sieg, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence1990, xiv + 297 pp. - Douglas K. Brown. Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. Pp. 39–50. - Kostas Hatzikiriakou and Stephen G. Simpson. WKL0 and orderings of countable abelian groups. Pp. 177–180. - Jeffry L. Hirst. Marriage theorems and reverse mathematics. Pp. 181–196. - Xiaokang Yu. Radon–Nikodym theorem is equivalent to arithmetical comprehension. Pp. 289–297. - Fernando Ferreira. Polynomial time computable arithmetic. Pp. 137–156. - Wilfried Buchholz and Wilfried Sieg. A note on polynomial time computable arithmetic. Pp. 51–55. - Samuel R. Buss. Axiomatizations and conservation results for fragments of bounded arithmetic. Pp. 57–84. - Gaisi Takeuti. Sharply bounded arithmetic and the function a – 1. Pp. 281–288. - Peter G. Clote. A smash-based hierarchy between PTIME and PSPACE (preliminary version). Pp. 85–100. - Solomon Feferman. Polymorphic typed lambda-calculi in a type-free axiomatic framework. Pp. 101–136 - Michael Beeson. Some theories conservative over intuitionistic arithmetic. Pp. 1–15.

1996 ◽  
Vol 61 (2) ◽  
pp. 697-699
Author(s):  
Jörg Hudelmaier
Keyword(s):  
Polynomial Time ◽  
Metric Space ◽  
American Mathematical Society ◽  
Bounded Arithmetic ◽  
Preliminary Version ◽  
Order Arithmetic ◽  
Complete Separable Metric Space ◽  
Separable Metric Space ◽  
Intuitionistic Arithmetic ◽  
Nikodym Theorem
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