The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “Extension of functions with small oscillation”

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

Similarity:

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β ( f ) ω ξ · ω ξ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

Similarity:

Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family f α : X Y ( α < ω ) of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set α < ω : f α ( x ) f ( x ) is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of Σ η sets which can be interesting in its own right.

Rudin-like sets and hereditary families of compact sets

Étienne Matheron, Miroslav Zelený (2005)

Fundamenta Mathematicae

Similarity:

We show that a comeager Π₁¹ hereditary family of compact sets must have a dense G δ subfamily which is also hereditary. Using this, we prove an “abstract” result which implies the existence of independent ℳ ₀-sets, the meagerness of ₀-sets with the property of Baire, and generalizations of some classical results of Mycielski. Finally, we also give some natural examples of true F σ δ sets.

Insertion of a Contra-Baire- 1 (Baire- . 5 ) Function

Majid Mirmiran (2019)

Communications in Mathematics

Similarity:

Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire- . 5 function between two comparable real-valued functions on the topological spaces that F σ -kernel of sets are F σ -sets.

Baire one functions and their sets of discontinuity

Jonald P. Fenecios, Emmanuel A. Cabral, Abraham P. Racca (2016)

Mathematica Bohemica

Similarity:

A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function f : is of the first Baire class if and only if for each ϵ > 0 there is a sequence of closed sets { C n } n = 1 such that D f = n = 1 C n and ω f ( C n ) < ϵ for each n where ω f ( C n ) = sup { | f ( x ) - f ( y ) | : x , y C n } and D f denotes the set of points of discontinuity of f . The proof of the main theorem is based on a recent ϵ - δ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications...

Distances to spaces of affine Baire-one functions

Jiří Spurný (2010)

Studia Mathematica

Similarity:

Let E be a Banach space and let ( B E * ) and ( B E * ) denote the space of all Baire-one and affine Baire-one functions on the dual unit ball B E * , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between d i s t ( f , ( B E * ) ) and d i s t ( f , ( B E * ) ) , where f is an affine function on B E * . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

A remark on functions continuous on all lines

Luděk Zajíček (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that each linearly continuous function f on n (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016). The same result holds also for f on an arbitrary Banach space X , if f has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such f on a separable X is continuous at all points outside a first category set which is also null in any usual...

Baire classes of complex L 1 -preduals

Pavel Ludvík, Jiří Spurný (2015)

Czechoslovak Mathematical Journal

Similarity:

Let X be a complex L 1 -predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire- α functions on the set ext B X * of the extreme points of the dual unit ball B X * to the whole unit ball B X * . As a corollary we show that, given α [ 1 , ω 1 ) , the intrinsic α -th Baire class of X can be identified with the space of bounded homogeneous Baire- α functions on the set ext B X * when ext B X * satisfies certain topological assumptions. The paper is intended to be a complex counterpart to...

On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

Similarity:

We study the Borel reducibility of Borel equivalence relations on the generalized Baire space κ κ for an uncountable κ with κ < κ = κ . The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.

Operators on the stopping time space

Dimitris Apatsidis (2015)

Studia Mathematica

Similarity:

Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure μ x * * on the Cantor set. We use the measures μ x * * : x * * ( S ¹ ) to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if μ T * * ( x * * ) : x * * ( S ¹ ) is non-separable as a subset of ( 2 ) .

Generic power series on subsets of the unit disk

Balázs Maga, Péter Maga (2022)

Czechoslovak Mathematical Journal

Similarity:

We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f ( U ) is a dense set of . As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property...

Consistency of the Silver dichotomy in generalised Baire space

Sy-David Friedman (2014)

Fundamenta Mathematicae

Similarity:

Silver’s fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space κ κ for a regular uncountable κ fails in Gödel’s L, even for κ-Borel equivalence relations. We show here that Silver’s dichotomy for κ-Borel equivalence relations in κ κ for uncountable regular κ is however consistent (with...

Decomposing Borel functions using the Shore-Slaman join theorem

Takayuki Kihara (2015)

Fundamenta Mathematicae

Similarity:

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each F σ set under that function is again F σ . Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

Diagonals of separately continuous functions of n variables with values in strongly σ -metrizable spaces

Olena Karlova, Volodymyr Mykhaylyuk, Oleksandr Sobchuk (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove the result on Baire classification of mappings f : X × Y Z which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where X is a P P -space, Y is a topological space and Z is a strongly σ -metrizable space with additional properties. We show that for any topological space X , special equiconnected space Z and a mapping g : X Z of the ( n - 1 ) -th Baire class there exists a strongly separately continuous mapping f : X n Z with the diagonal g . For wide classes...

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

For a multidimensional control problem ( P ) K involving controls u L , we construct a dual problem ( D ) K in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of ( L ) * . For this purpose, we add to ( P ) K a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

Boaz Tsaban, Lubomyr Zdomsky (2012)

Journal of the European Mathematical Society

Similarity:

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel’skii α 1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C p ( X ) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our...

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

Similarity:

We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel...

Sharkovskiĭ's theorem holds for some discontinuous functions

Piotr Szuca (2003)

Fundamenta Mathematicae

Similarity:

We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected G δ graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected G δ graph.

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

Similarity:

If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball B E * that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of B E * , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...