Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T14:09:52.693Z Has data issue: false hasContentIssue false
  • English
  • Français

Insertion of a measurable function

Part of: Maps and general types of spaces defined by maps Classical measure theory Real functions

Published online by Cambridge University Press:  09 April 2009

Wesley Kotzé  and
Tomasz Kubiak
Wesley Kotzé
Affiliation:
Department of Mathematics, (Pure and Applied), Rhodes University, Grahamstown 6140, South Africa
Tomasz Kubiak
Affiliation:
Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some theorems on the existence of continuous real-valued functions on a topological space (for example, insertion, extension, and separation theorems) can be proved without involving uncountable unions of open sets. In particular, it is shown that well-known characterizations of normality (for example the Katětov-Tong insertion theorem, the Tietze extension theorem, Urysohn's lemma) are characterizations of normal σ-rings. Likewise, similar theorems about extremally disconnected spaces are true for σ-rings of a certain type. This σ-ring approach leads to general results on the existence of functions of class α.

Keywords

real-valued functionσ-ringmeasurable functionclass α functioninsertionextensionseparationperfect spacenormalextremally disconnected.

MSC classification

Secondary: 54C50: Special sets defined by functions 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 54C20: Extension of maps 54C45: $C$- and $C^*$-embedding 26A21: Classification of real functions; Baire classification of sets and functions 54C30: Real-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 3 , December 1994 , pp. 295 - 304
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Blair, R. L., ‘Extension of Lebesgue sets and of real-valued functions’, Czechoslovak Math. J. 31 (1981), 63–74.CrossRefGoogle Scholar
[2]Engelking, R., General topology (PWN, Warsaw, 1977).Google Scholar
[3]Gillman, L. and Jerison, M., Rings of continuous functions (Springer, Berlin, 1976).Google Scholar
[4]Katětov, M., ‘On real-valued functions in topological spaces’, Fund. Math. 38 (1951), 85–91CrossRefGoogle Scholar
‘Correction’, Fund. Math. 40 (1953), 203–205.Google Scholar
[5]Kotzé, W., ‘Functions of class one and well-behaved spaces’, Rend. Mat. Appl. 4 (1984), 139–156.Google Scholar
[6]Kubiak, T., ‘Normality versus extremal disconnectedness’, preprint.Google Scholar
[7]Kuratowski, K., Topology, Vol. 1 (Academic Press, New York, 1966).Google Scholar
[8]Kutateladze, S. S., Foundations of functional analysis (Nauka, Novosibirsk, 1983) (in Russian).Google Scholar
[9]Lane, E. P., ‘A sufficient condition for the insertion of a continuous function’, Proc. Amer. Math. Soc. 49 (1975), 90–94.CrossRefGoogle Scholar
[10]Lane, E. P., ‘Insertion of a continuous function’, Topology Proc. 4 (1979), 463–478.Google Scholar
[11]Mrówka, S., ‘On some approximation theorems’, Nieuw Arch. Wisk. 16 (1968), 94–111.Google Scholar
[12]Mustafa, I., ‘Insertion of Darboux Baire one function’, Real Anal. Exchange 12 (1986/1987), 458–467.CrossRefGoogle Scholar
[13]Preiss, D. and Vilimovský, J., ‘In-between theorems in uniform spaces’, Trans. Amer. Math. Soc. 261 (1980), 483–501.CrossRefGoogle Scholar
[14]Sikorski, R., Real functions, Vol. 1 (PNW, Warsaw, 1958) (in Polish).Google Scholar
[15]Speed, T. P., ‘On rings of sets II. zero sets’, J. Austral. Math. Soc. Ser. A 16 (1973), 185–199.CrossRefGoogle Scholar
[16]Tong, H., ‘Some characterizations of normal and perfectly normal spaces’, Duke Math. J. 19 (1952), 289–292.CrossRefGoogle Scholar