scholarly journals Note on limits of simply continuous and cliquish functions

1994 ◽  
Vol 17 (3) ◽  
pp. 447-450 ◽  
Author(s):  
Janina Ewert
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Baire Property ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
Separable Metric Space

The main result of this paper is that any functionfdefined on a perfect Baire space(X,T)with values in a separable metric spaceYis cliquish (has the Baire property) iff it is a uniform (pointwise) limit of sequence{fn:n≥1}of simply continuous functions. This result is obtained by a change of a topology onXand showing that a functionf:(X,T)→Yis cliquish (has the Baire property) iff it is of the Baire class 1 (class 2) with respect to the new topology.

scholarly journals Baire and automata

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Pierre Simonnet ◽  
Benoit Cagnard
Keyword(s):  
Computer Science ◽  
Rational Function ◽  
Continuous Functions ◽  
Transfinite Induction ◽  
Baire Class ◽  
Class 1 ◽  
Pointwise Limit ◽  
International Audience ◽  
Büchi Automaton

International audience In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.

A theorem about countable decomposability

1982 ◽  
Vol 91 (3) ◽  
pp. 457-458 ◽  
Author(s):  
Roy O. Davies ◽  
Claude Tricot
Keyword(s):  
Topological Space ◽  
Metric Space ◽  
Continuous Functions ◽  
Perfect Set ◽  
Baire Function ◽  
Separable Metric Space

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

Separate approximate continuity implies measurability

1973 ◽  
Vol 73 (3) ◽  
pp. 461-465 ◽  
Author(s):  
Roy O. Davies
Keyword(s):  
Continuum Hypothesis ◽  
Measurable Cardinal ◽  
Continuous Functions ◽  
Affirmative Answer ◽  
Baire Class ◽  
Continuous Real Function ◽  
Approximate Continuity ◽  
Class 1 ◽  
Measure Spaces ◽  
First Baire Class

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Mišik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

A note on ρ-upper continuous functions

2009 ◽  
Vol 44 (1) ◽  
pp. 153-158
Author(s):  
Stanisław Kowalczyk ◽  
Katarzyna Nowakowska
Keyword(s):  
Continuous Functions ◽  
Baire Class ◽  
Class 1 ◽  
Upper Continuous

Abstract In the present paper, we introduce the notion of classes of ρ-upper continuous functions. We show that ρ-upper continuous functions are Lebesgue measurable and, for ρ < 1/2 , may not belong to Baire class 1. We also prove that a function with Denjoy property can be non-measurable.

A very discontinuous Borel function

1993 ◽  
Vol 58 (4) ◽  
pp. 1268-1283 ◽  
Author(s):  
Juris Steprāns
Keyword(s):  
Borel Function ◽  
Continuous Functions ◽  
Baire Class ◽  
Cardinal Invariant ◽  
Class 1

AbstractIt is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.

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 Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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