Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T00:16:00.504Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological vitali measure spaces

Published online by Cambridge University Press:  17 April 2009

D.N. Sarkhel  and
T. Chakraborti
D.N. Sarkhel
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India.
T. Chakraborti
Affiliation:
Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India.
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.

The properties of Lebesgue outer measures embodied in the Vitali covering theorem, the Vitali-Carathéodory theorem, the Lusin theorem, the density theorem, outer regularity and inner regularity, and the relation between measurability and approximate continuity are studied in a general abstract space, called a topological Vitali measure space. The main theme is the mutual equivalence of these properties.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 32 , Issue 2 , October 1985 , pp. 225 - 249
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Comfort, W.W. and Gordon, H., “Vitali's theorem for invariant measures”, Trans. Amer. Math. Soc. 99 (1961), 83–90.Google Scholar
[2]Eames, W., “A local property of measurable sets”, Canad. J. Math. 12 (1960), 632–640.CrossRefGoogle Scholar
[3]Freilich, G., “Gauges and their densities”, Trans. Amer. Math. Soc. 122 (1966), 153–162.CrossRefGoogle Scholar
[4]Goffman, C. and Waterman, D., “Ipproximately continuous transformations”, Proc. Amer. Math. Soc. 12 (1961), 116–121.CrossRefGoogle Scholar
[5]Hahn, H. and Rosenthal, A., Set functions (University of New Mexico Press, Albuquerque, 1948).Google Scholar
[6]Halmos, P.R., Measure theory (Van Nostrand, New York, 1950).CrossRefGoogle Scholar
[7]Hayes, C.A. Jr. and Pauc, C.Y., “Full individual and class differentiation theorems in their relations to halo and Vitali properties”, Canad. J. Math. 7 (1955), 221–274.CrossRefGoogle Scholar
[8]Kelley, J.L., General topology (Van Nostrand, New York, 1955).Google Scholar
[9]Lahiri, B.K., “Density and approximate continuity in topological groups”, J. Indian Math. Soc. 41 (1977), 129–141.Google Scholar
[10]Morse, A.P., “A theory of covering and differentiation”, Trans. Amer. Math. Soc. 55 (1944), 205–235.CrossRefGoogle Scholar
[11]Saks, S., Theory of the integral (2nd rev. ed., PWN, Warsaw, 1937; reprint, Dover, New York, 1964).Google Scholar
[12]Sarkhel, D.N., “Vitali covering theorem in topological spaces”, Bull. Cal. Math. Soc. 66 (1974), 101–106.Google Scholar
[13]Sarkhel, D.N., “A generalization of the Vitali covering theorem”, Fund. Math. 97 (1977), 151–156.CrossRefGoogle Scholar
[14]Sion, M., “Approximate continuity and differentiation”, Canad. J. Math. 14 (1962), 467–475.CrossRefGoogle Scholar
[15]Wage, M.L., “A generalization of Lusin's theorem”, Proc. Amer. Math. Soc. 52 (1975), 327–332.Google Scholar