CLASSIFICATIONS OF BOREL MEASURABLE FUNCTIONS

We introduce the new concept of pointwise measurability. It is shown in this paper that a measurable function is measurable at each point and that for a large class of topological spaces the converse also holds. Moreover it can be seen that a function which is continuous at a point is Borel-measurable at this point too. Furthermore the set of measurability points is considered. If the range space is a $\sigma$-compact metric space, then this set is a $G_{\delta}$-set; if the range space is only a Polish space this is in general not true any longer.

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On the difference property of Borel measurable functions

Fundamenta Mathematicae ◽

10.4064/fm208-1-4 ◽

2010 ◽

Vol 208 (1) ◽

pp. 57-73

Author(s):

Hiroshi Fujita ◽

Tamás Mátrai

Keyword(s):

Difference Property ◽

Measurable Functions ◽

The Difference ◽

Borel Measurable

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On the Dirichlet Problem in the Axiomatic Theory of Harmonic Functions

Nagoya Mathematical Journal ◽

10.1017/s0027763000011193 ◽

1963 ◽

Vol 23 ◽

pp. 73-96 ◽

Cited By ~ 6

Author(s):

N. Boboc ◽

C. Constantinescu ◽

A. Cornea

Keyword(s):

Dirichlet Problem ◽

Harmonic Functions ◽

Riemann Surfaces ◽

Minimum Principle ◽

Continuous Functions ◽

Axiomatic Theory ◽

Measurable Functions ◽

Bounded Functions ◽

Compact Riemann Surfaces ◽

Borel Measurable

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.

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A Property of Borel Measurable Functions and Extendable Functions

Real Analysis Exchange ◽

10.2307/44151833 ◽

1987 ◽

Vol 13 (1) ◽

pp. 11

Author(s):

Gibson

Keyword(s):

Measurable Functions ◽

Borel Measurable

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Infinite products of Borel measurable functions

Topology and its Applications ◽

10.1016/j.topol.2007.04.027 ◽

2008 ◽

Vol 155 (17-18) ◽

pp. 1996-2000 ◽

Cited By ~ 1

Author(s):

Katarzyna Chmielewska ◽

Aleksander Maliszewski

Keyword(s):

Infinite Products ◽

Measurable Functions ◽

Borel Measurable

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Some theorems on Borel-measurable functions

Colloquium Mathematicum ◽

10.4064/cm-10-2-261-266 ◽

1963 ◽

Vol 10 (2) ◽

pp. 261-266 ◽

Cited By ~ 1

Author(s):

G. Lederer

Keyword(s):

Measurable Functions ◽

Borel Measurable

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On a Lindenbaum Composition Theorem

Tatra Mountains Mathematical Publications ◽

10.2478/tmmp-2019-0025 ◽

2019 ◽

Vol 74 (1) ◽

pp. 145-158

Author(s):

Jaroslav Šupina ◽

Dávid Uhrik

Keyword(s):

Topological Space ◽

Real Line ◽

Topological Spaces ◽

The Real ◽

Measurable Functions ◽

Borel Measurable ◽

Composition Theorem ◽

Perfectly Normal

Abstract We discuss several questions about Borel measurable functions on a topological space. We show that two Lindenbaum composition theorems [Lindenbaum, A. Sur les superpositions des fonctions représentables analytiquement, Fund. Math. 23 (1934), 15–37] proved for the real line hold in perfectly normal topological space as well. As an application, we extend a characterization of a certain class of topological spaces with hereditary Jayne-Rogers property for perfectly normal topological space. Finally, we pose an interesting question about lower and upper Δ02-measurable functions.

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