An effective selection theorem

1982 ◽  
Vol 47 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Ashok Maitra
Keyword(s):  
Polish Space ◽  
Local Version ◽  
Compact Set ◽  
Selection Theorem ◽  
Analytic Subset ◽  
Effective Version ◽  
Basis Theorem ◽  
Measurable Selector ◽  
Effective Selection ◽  
Borel Measurable

A recent result of J.P. Burgess [1] states:Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.

On the existence of an analytic set meeting each compact set in a Borel set

1978 ◽  
Vol 84 (1) ◽  
pp. 5-10 ◽  
Author(s):  
A. R. D. Mathias ◽  
A. J. Ostaszewski ◽  
M. Talagrand
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Continuum Hypothesis ◽  
Compact Set ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Counter Example ◽  
The Axiom Of Choice ◽  
The Continuum

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

1.—On Measurable Selections

1974 ◽  
Vol 72 (1) ◽  
pp. 1-7 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Measure Space ◽  
Hausdorff Space ◽  
Polish Space ◽  
Measurable Selection ◽  
Selection Theorem ◽  
Polish Spaces ◽  
Measurable Multifunction ◽  
Selection Theorems ◽  
Suitable Measure ◽  
Separable Metric Space

SynopsisMeasurable selection theorems are proved, for a compact-valued measurable multifunction into a Hausdorff space that is the continuous image of a separable metric space, and for a closed-valued measurable multifunction from a suitable measure space to a regular Souslin space. The connection between Polish spaces and certain subsets of the real line is related to a measurable selection theorem for multifunctions into a Polish space.

scholarly journals Omega-Polish spaces and measurable selections

1977 ◽  
Vol 23 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Le Van Tu
Keyword(s):  
Large Class ◽  
Banach Spaces ◽  
Polish Space ◽  
Measurable Space ◽  
Selection Theorem ◽  
Polish Spaces ◽  
Measurable Selections

AbstractIn this paper, the author introduces the notion of Ω-Polish spaces (which includes the Polish spaces and a large class of Banach spaces) and extends Castaing's selection theorem (1966) for closed-valued measurable thin multifunctions from a measurable space into an Ω-Polish space. He also extends Robertson's theorem (1974) in the same way.

scholarly journals Pointwise measurable functions

2004 ◽  
Vol 95 (2) ◽  
pp. 305
Author(s):  
Herman Render ◽  
Lothar Rogge
Keyword(s):  
Large Class ◽  
Metric Space ◽  
Measurable Function ◽  
Polish Space ◽  
Topological Spaces ◽  
Range Space ◽  
Compact Metric Space ◽  
Delta Set ◽  
Measurable Functions ◽  
Borel Measurable

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

scholarly journals On continuity and selections of multifunctions

2002 ◽  
Vol 65 (3) ◽  
pp. 407-422
Author(s):  
Pandelis Dodos
Keyword(s):  
Banach Space ◽  
Polish Space ◽  
Lower Semicontinuous ◽  
Selection Theorem ◽  
Weakly Compact ◽  
Compact Subsets

The notions of a Baire-1 and a weak Baire-1 multifunction are defined and a striking analogy between Baire-1 multifunctions and classical Baire-1 functions is established. A selection theorem is presented which asserts that if X is a metrisable space, Y a Polish space and F: X → 2Y/{∅} a closed-valued, weak Baire-1 multifunction, then F admits a Baire-1 selection. Using the machinery developed we prove that if X is a Banach space with separable dual, then every weak* usco, defined on a completely metrisable space Z, which values are weakly* compact subsets of the dual, is norm lower semicontinuous on a dense Gδ set.

Tree structures associated to a family of functions

2005 ◽  
Vol 70 (3) ◽  
pp. 681-695 ◽  
Author(s):  
Spiros A. Argyros ◽  
Pandelis Dodos ◽  
Vassilis Kanellopoulos
Keyword(s):  
Polish Space ◽  
Uniform Boundedness ◽  
Index Theorem ◽  
Continuous Functions ◽  
Baire Class ◽  
Compact Set ◽  
Limit Function ◽  
Tree Structures ◽  
Pointwise Limit ◽  
First Baire Class

The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform boundedness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta and we shall denote by ∝(f) the separation rank of a real-valued function f in B1(X), with X a Polish space. Notice that in [3], Bourgain has provided a positive answer to this problem in the case of K satisfying with X a compact metric space. The key ingredient in Bourgain's approach is that whenever a sequence of continuous functions pointwise converges to a function f, then the possible discontinuities of the limit function reflect a local ℓ1-structure to the sequence (fn)n. More precisely the complexity of this ℓ1-structure increases as the complexity of the discontinuities of f does. This fruitful idea was extensively studied by several authors (c.f. [5], [7], [8]) and for an exposition of the related results we refer to [1]. It is worth mentioning that A.S. Kechris and A. Louveau have invented the rank rND(f) which permits the link between the c0-structure of a sequence (fn)n of uniformly bounded continuous functions and the discontinuities of its pointwise limit. Rosenthal's c0-theorem [11] and the c0-index theorem [2] are consequences of this interaction.Passing to the case where either (fn)n are not continuous or X is a non-compact Polish space, this nice interaction is completely lost.

scholarly journals Turing degrees in Polish spaces and decomposability of Borel functions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050021
Author(s):  
Vassilios Gregoriades ◽  
Takayuki Kihara ◽  
Keng Meng Ng
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Analytic Subset ◽  
Polish Spaces ◽  
Negative Results ◽  
Borel Functions ◽  
Important Open Problem ◽  
Descriptive Set

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

scholarly journals BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON HARDY SPACES

2008 ◽  
Vol 51 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Marco M. Peloso ◽  
Silvia Secco
Keyword(s):  
Hardy Spaces ◽  
Bounded Operator ◽  
Integral Operators ◽  
Fourier Integral ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Compact Set ◽  
Radon Transforms ◽  
Bounded Geometry ◽  
Local Hardy Spaces

AbstractFor $0\ltp\le1$, let $h^p(\mathbb{R}^n)$ denote the local Hardy space. Let $\mathcal{F}$ be a Fourier integral operator defined by the oscillatory integral$$ \mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi, $$where $\phi$ is a $\mathcal{C}^\infty$ non-degenerate real phase function, and $b$ is a symbol of order $\mu$ and type $(\rho,1-\rho)$, $\sfrac12\lt\rho\le1$, vanishing for $x$ outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$ and $\mu\le-(n-1)(1/p-1/2)$ then $\mathcal{F}$ initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$ extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$ and $\mu$ is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$ spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces.Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry.

The topological Vaught's conjecture and minimal counterexamples

1994 ◽  
Vol 59 (3) ◽  
pp. 757-784 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Topological Group ◽  
Polish Space ◽  
Sufficient Conditions ◽  
Usual Model ◽  
Borel Set ◽  
Isomorphism Classes ◽  
Perfect Set ◽  
Minimal Counterexample ◽  
Necessary And Sufficient ◽  
Borel Measurable

Let G be a Polish topological group, let X be a Polish space, let J: G × X → X be a Borel-measurable action of G on X, and let A ⊂ X be a Borel set which is invariant with respect to J, i.e., a Borel set of orbits. The following statement, or various equivalent versions of it, is known as the Topological Vaught's Conjecture.Let (G, X, J, A) be as above. Either A contains only countably many orbits, or else, A contains perfectly many orbits.We say that A contains perfectly many orbits if there is a perfect set P ⊂ A such that no two elements of P are in the same orbit. (Assuming ¬CH, A contains perfectly many orbits iff it contains 2ℵ0 orbits.) The Topological Vaught's Conjecture implies the usual, model theoretic, Vaught's Conjecture for Lω1ω, since the isomorphism classes are the orbits of an action of the group of permutations of ω; we give details in §0. The “Borel” assumption cannot be weakened for either A or J.Given a Borel-measurable Polish action (G,X,J) and an invariant Borel set B ⊂ X, we say that B is a minimal counterexample if (G,X,J,B) is a counterexample to the Topological Vaught's Conjecture and for every invariant Borel C ⊂ B, either C or B\C contains only countably many orbits. This paper is concerned with counterexamples to the Topological Vaught's Conjecture (of course, there may not be any), and in particular, with minimal counterexamples. First, there is a theorem on the existence of minimal counterexamples. This theorem was known for the model theoretic case (it is due to Harnik and Makkai), and is here generalized to arbitrary Borel-measurable Polish actions. Second, we study the properties of minimal counterexamples. We give two different necessary and sufficient conditions for a counterexample to be minimal, as well as some consequences of minimality. Some of these results are proved assuming determinacy axioms.This second part seems to be new even in the model theoretic case.

State-cathedral nature of the first-rate Kiev

2000 ◽  
pp. 60-67
Author(s):  
M. M. Nikitenko
Keyword(s):  
World History ◽  
Cultural Influences ◽  
Private Life ◽  
Local Version ◽  
Russian Society ◽  
Public And Private ◽  
Main Center

The inclusion of Eastern Slavs in the sphere of religious and cultural influences of Byzantium was a tremendous event both in national and in world history. Since then, the main center of the culture of Kievan Rus, incorporating a complex of ideas and functions of the spiritual, public and private life of ancient Russian society, became the Eastern Christian temple in its local version

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