Functions of the 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.

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Separate approximate continuity implies measurability

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100077033 ◽

1973 ◽

Vol 73 (3) ◽

pp. 461-465 ◽

Cited By ~ 6

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.

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The cardinality of first Baire class

Strange Functions in Real Analysis ◽

10.1201/9781315154473-10 ◽

2017 ◽

pp. 143-152

Author(s):

Gilbert W. Bassett Jr. ◽

Roger Koenker

Keyword(s):

Baire Class ◽

First Baire Class

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Functions of the first Baire class with values in metrizable spaces

Ukrainian Mathematical Journal ◽

10.1007/s11253-006-0089-2 ◽

2006 ◽

Vol 58 (4) ◽

pp. 640-644 ◽

Cited By ~ 5

Author(s):

O. O. Karlova ◽

V. V. Mykhailyuk

Keyword(s):

Baire Class ◽

First Baire Class ◽

Metrizable Spaces

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Two Effective Properties of ω-Rational Functions

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500283 ◽

2021 ◽

pp. 1-20

Author(s):

Olivier Finkel

Keyword(s):

Effective Properties ◽

Rational Functions ◽

Baire Class ◽

Infinite Words ◽

Finite State ◽

First Baire Class ◽

Büchi Automaton

We prove two new effective properties of rational functions over infinite words which are realized by finite state Büchi transducers. Firstly, for each such function [Formula: see text], one can construct a deterministic Büchi automaton [Formula: see text] accepting a dense [Formula: see text]-subset of [Formula: see text] such that the restriction of [Formula: see text] to [Formula: see text] is continuous. Secondly, we give a new proof of the decidability of the first Baire class for synchronous [Formula: see text]-rational functions from which we get an extension of this result involving the notion of Wadge classes of regular [Formula: see text]-languages.

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