Descriptive Set Theory

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit interval I = [0,1] into itself. More recently, the notion of Baire class-1 function turned out to also be important in some areas of functional analysis (see [R3]). For example, Odell and Rosenthal [OR] showed that the double dual of a separable Banach space E with the weak* topology consists only of Baire class-1 functions defined on the unit ball of E* if and only if the space E contains no subspace isomorphic to ?1. This resulted in a renewed interest in this class of spaces. For example, building on the work of Rosenthal [R2], Bourgain, FYemlin and Talagrand [BFT] proved analogues of the two theorems of Helly for the whole first Baire class. Using their results Godefroy [Go] showed that this class of spaces enjoys some interesting permanence properties. For example, if a compact space K is representable as a compact set of Baire class-1 functions, then so is P(K), the space of all Radon probability measures on K with the weak* topology. Some further permanence properties of this class of spaces were obtained by Marciszewski ([Ml], [M2]) and an excellent survey of the early results is given by R. Pol [Po2]. Our paper is an attempt towards a fine structure theory of compact subsets of first Baire class. The first result that we give is a positive answer to a natural question one usually asks in such a context.

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Compact subsets of the first Baire class

Internal characterization of fragmentable spaces

This chapter discusses renorming a Banach space that consists of replacing the given norm that is usually provided by the very definition of the space, by another norm that may have better (or sometimes worse) properties of convexity or smoothness, or both. This operation is by nature geometric. The geometric intuition is frequently useful and a few pictures might help in understanding certain proofs. The analytical point of view is of course necessary and proofs usually rely on careful computations. Computing in Banach spaces is quite difficult, because certain spaces have no basis and, therefore, there is no hope to use co-ordinates in calculations. Most results in renorming theory rely on topological assumptions. Norms that enjoy good properties of convexity and/or smoothness can be computed under natural (and optimal) topological assumptions. These norms are frequently obtained by duality arguments and they are tightly connected with the linear structure.

Chapter 18 – Renormings of Banach Spaces

We prove a result concerning separate and joint continuity that contains a celebrated result of Namioka. In 1974 Namioka proved a result somewhat stronger than the following: THEOREM I [N1]. Let B be a Cech-complete topological space, K a compact Hausdorff space and f: B x K -* M a separately continuous function into the metric space M. Then there exists a dense G6 subset G of B such that f is (jointly) continuous at each point of G x K. There is an obvious reformulation of this result. Suppose f, B and K are as above and C(K) denotes the Banach space of continuous functions on K. Then the function f induces a function h: B -* C(K) defined by h(b) = f(b, ) and h is continuous into C(K) when C(K) has the simple topology. Our own formulation of Namioka's result, with a considerably different proof, is the following: THEOREM II [S2]. Suppose B is a Oech-complete space and f: B -X is a function into the Banach space X such that for any x* in X* that is an extreme point of the unit ball of X* we have that x* o f is continuous. Then f is normcontinuous at each point of a dense G6 subset G of B. We postpone until the end of the paper a proof of Theorem II. We shall say that such a function f is a(X, ext) continuous. We shall show that a number of recent results concerning separate continuity are consequences of this theorem. Two natural problems which have been investigated by various authors are the following: (1) characterize those Banach spaces X such that for any Baire space B and any f: B -* X that is weakly continuous we have that f is norm-continuous at each point of a dense G6 set; (2) characterize those Baire spaces for which Namioka's theorem is always valid. We show that in all of the known cases we need only assume in (1) that f is continuous with respect to the extremal topology. We reduce a number of results concerning problems (1) and (2) to Theorem I (see Theorem 3 below). Only Hausdorff topological spaces will be considered. For background material in topology, see [E and B]. Let T be a topological space. We shall say that T is k-analytic, written ka, if there exist a tech-complete and Lindel6f space C and an onto continuous function Received by the editors January 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B99; Secondary 26B05. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

/pdf/generalizations-of-a-theorem-of-namioka-2c8nd8yxyi.pdf

Generalizations of a theorem of Namioka

We first prove that given any analytic filter F on ω the set of all functions f on 2 which can be represented as the pointwise limit relative to F of some sequence (fn)n∈ω of continuous functions (f = limF fn), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of F . We discuss several structural properties of this rank. For example, we prove that any free Π4 filter is of rank 1. The idea of associating to a given filter F a class of functions is due to Katětov (see [5] and [6]). In this work we restrict our study to classes of Borel functions on Polish spaces and we shall assume the reader to be familiar with Descriptive Set Theory in this context. Nevertheless to avoid any ambiguity we shall recall explicitly in this introduction a number of elementary notions. Descriptive pointclasses. Informally speaking, a descriptive pointclass Γ is a general assignment to any Polish space X of a family Γ(X) of subsets of X with the property that for any continuous mapping φ : X → Y between two Polish spaces X and Y , if A ∈ Γ(Y ) then φ−1(A) ∈ Γ(X). More formally, Γ should be viewed as a formula in the language of Set Theory, and Γ(X) the set obtained by relativization of this formula to the structure induced by the Polish topology on X. Still we shall very often refer to Γ(X) as a “class of sets on X”. Typical examples of such classes are the (additive, multiplicative, ambiguous) Borel classes Σξ , Π 0 ξ , ∆ 0 ξ or the classes ∆ 1 1, Σ 1 1, Π1 (of Borel, analytic, coanalytic sets). Throughout, by a function we shall always mean a real-valued mapping. Descriptive classes of functions. In the same way, a descriptive class D of functions is a general assignment to any Polish space X of a family D(X) 2000 Mathematics Subject Classification: Primary 03E15; Secondary 03E45, 54A20,

/pdf/filter-descriptive-classes-of-borel-functions-5c9ini5mpb.pdf

Filter descriptive classes of Borel functions

Introduction A tree representation for Borel sets A double-tree representation for Borel sets Two applications of the tree representation Borel liftings of Borel sets More consequences and reverse results Proof of the main result Bibliography Index.

/pdf/borel-liftings-of-borel-sets-some-decidable-and-undecidable-307fiazk2g.pdf

Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

© Annales de l’institut Fourier, 1987, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/ensembles-boreliens-d-unicite-et-d-unicite-au-sens-large-4rk144gkl6.pdf

Ensembles boréliens d'unicité et d'unicité au sens large

Effective properties in compact sets of Borel functions

Compact covering and game determinacy

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