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.

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.

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 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.

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.

scholarly journals Transfinite sequences of continuous and Baire class 1 functions

2002 ◽  
Vol 131 (8) ◽  
pp. 2453-2457 ◽  
Author(s):  
Márton Elekes ◽  
Kenneth Kunen
Keyword(s):  
Baire Class ◽  
Class 1

scholarly journals A Classification of Baire Class 1 Functions

1990 ◽  
Vol 318 (1) ◽  
pp. 209 ◽  
Author(s):  
A. S. Kechris ◽  
A. Louveau
Keyword(s):  
Baire Class ◽  
Class 1

Baire class 1

2001 ◽  
pp. 13-20
Author(s):  
Casper Goffman ◽  
Togo Nishiura ◽  
Daniel Waterman
Keyword(s):  
Baire Class ◽  
Class 1

Analytic Ideals

1996 ◽  
Vol 2 (3) ◽  
pp. 339-348 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Partial Order ◽  
Polish Space ◽  
Equivalence Relations ◽  
Baire Class ◽  
Maximal Ideals ◽  
Borel Ideal ◽  
Class 1 ◽  
Long Time ◽  
Analytic Ideals ◽  
Substantial Interest

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters (or, dually, maximal ideals). There is also a substantial interest in nicely definable (Borel, analytic) ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space (Rosenthal compacta), see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P(ω): X ≥I Y iff X \ Y ϵ I ([16]) and the partial order (I, ⊂)([18]).2. Boolean algebras of the form P(ω)/I and their automorphisms ([6, 5, 19, 20]).3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I ([4, 14, 15,9]).In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3.

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