Tree structures associated to a family of functions

Author:

Argyros Spiros A.,Dodos Pandelis,Kanellopoulos Vassilis

Abstract

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.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Reference13 articles.

1. Pointwise Compact Sets of Baire-Measurable Functions

2. Topics in Topology

3. On certain classes of Baire-1 functions with applications to Banach space theory

4. On convergent sequences of continuous functions;Bourgain;Bulletin de la Société Mathématique de Belgique,1980

5. Descriptive Set Theory and Banach Spaces

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1. Descriptive Aspects of Rosenthal Compacta;Recent Progress in General Topology III;2013-12-12

2. The quantitative difference between countable compactness and compactness;Journal of Mathematical Analysis and Applications;2008-07

3. A classification of separable Rosenthal compacta and its applications;Dissertationes Mathematicae;2008

4. Spaces of functions with countably many discontinuities;Israel Journal of Mathematics;2007-03

5. Codings of separable compact subsets of the first Baire class;Annals of Pure and Applied Logic;2006-10

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