scholarly journals Functions of the first Baire class

2018 ◽  
Vol 24 (4) ◽  
pp. 462-464
Author(s):  
Raphaël Carroy
Keyword(s):  
Baire Class ◽  
First Baire Class

Lexicographic products as compact spaces of the first Baire class

2019 ◽  
Vol 267 ◽  
pp. 106871
Author(s):  
Antonio Avilés ◽  
Stevo Todorcevic
Keyword(s):  
Baire Class ◽  
Compact Spaces ◽  
First Baire Class

Functions of the First Baire Class

1988 ◽  
Vol s2-37 (3) ◽  
pp. 535-544 ◽  
Author(s):  
C. A. Rogers
Keyword(s):  
Baire Class ◽  
First Baire Class

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

scholarly journals Sifat Aljabar Banach Komutatif dan Elemen Identitas pada Kelas D(K)

CAUCHY ◽  
2013 ◽  
Vol 3 (1) ◽  
pp. 26
Author(s):  
Malahayati Malahayati
Keyword(s):  
Banach Algebra ◽  
Metric Spaces ◽  
Identity Element ◽  
Baire Class ◽  
Bounded Functions ◽  
First Baire Class ◽  
Semicontinuous Functions

The first Baire class of bounded functions on separable metric spaces K denoted by B1(k). One of the most important subclass of B1(k) is D(K), by D(K) is denoted the class of all functions on K which are differences of bounded semicontinuous functions. In this paper we proved that D(K) is abelian Banach algebra and identity element

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.

The cardinality of first Baire class

2017 ◽  
pp. 143-152
Author(s):  
Gilbert W. Bassett Jr. ◽  
Roger Koenker
Keyword(s):  
Baire Class ◽  
First Baire Class

Functions of the first Baire class with values in metrizable spaces

2006 ◽  
Vol 58 (4) ◽  
pp. 640-644 ◽  
Author(s):  
O. O. Karlova ◽  
V. V. Mykhailyuk
Keyword(s):  
Baire Class ◽  
First Baire Class ◽  
Metrizable Spaces
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