Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T10:35:47.487Z Has data issue: false hasContentIssue false
  • English
  • Français

A general measure decomposition theorem by means of the generalized wallman remainder

Part of: Classical measure theory

Published online by Cambridge University Press:  09 April 2009

George Bachman  and
Panagiotis D. Stratigos
George Bachman
Affiliation:
Department of Mathematics Polytechnic Institute of New YorkBrooklyn, New York 11201, U.S.A.
Panagiotis D. Stratigos
Affiliation:
Department of Mathematics Long Island UniversityBrooklyn, New York 11202, U.S.A.
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.

Let ℒ be a lattice of subsets of a set X. Let MR (ℒ) denote the set of all ℒ-regular (finitely additive) measures on the algebra generated by ℒ. Under the assumption that ℒ is disjunctive, in the first part of the paper, criteria are obtained for the σ-smoothness, τ-smoothness, and tightness of elements of MR(ℒ) in terms of the general Wallman remander. in the second part of the paper, various applications are given, and, in particular, extensions and refinements of the Yoside-Hewitt Decom position Theorem are obtained.

MSC classification

Secondary: 28A60: Measures on Boolean rings, measure algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 87 - 105
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Alexandroff, A. D., ‘Additive set functions in abstract spaces’, Math. Sb (N.S.) 9 51 (1941), 563–628.Google Scholar
[2]Bachman, G. and Sultan, A., ‘Regular lattice measures: mappings and spaces’, Pacific J. Math. 61 (1976), 291–321.CrossRefGoogle Scholar
[3]Bachman, G. and Sultan, A., ‘Extensions of regular lattice measures with topological applications’, J. Math. Anal. Appl. 57 (1977), 539–559.CrossRefGoogle Scholar
[4]Bachman, G. and Sultan, A., ‘Representations of linear functionals on spaces of continuous functions, repletions, and general measure externsions’, J. Math. Anal. Appl. 67 (1979), 277–293.CrossRefGoogle Scholar
[5]Bachman, G. and Sultan, A., ‘On regular extensions of measure’, Pacific J. Math., to appear.Google Scholar
[6]Banaschewski, B., ‘Über nulldimensionalle Raume’, Math. Nachr. 13 (1955), 129–140.CrossRefGoogle Scholar
[7]Gillman, L. and Jerrison, M., Rings of continuous functions (Van Nostrand, Princeton, N. J., (1960)).CrossRefGoogle Scholar
[8]Gould, G. G. and Mahowald, M., ‘Measures on completely regular spaces’, J. London Math. Soc. 37 (1962), 103–111.CrossRefGoogle Scholar
[9]Halmos, P., Measure theory (Van Nostrand, Toronto, New York, London, (1950)).CrossRefGoogle Scholar
[10]Knowles, J., ‘Measure on topological spaces’, Proc. London Math. Soc. 17 (1967), 139–156.CrossRefGoogle Scholar
[11]Lembcke, J., ‘Konservative Abbildungen und Fortsetzung regulärer Masse’, Z. Wahrschein-lichkeitsheorie und Verw. Gebiete 15 (1970), 57–96.CrossRefGoogle Scholar
[12]Moran, W., ‘Measures and mappings on topological spaces’, Proc. London Math. Soc. (3) 19 (1969), 493–508.CrossRefGoogle Scholar
[13]Sultan, A., ‘Measure compactification and representation’, Canad. J. Math. 30 (1978), 54–65.CrossRefGoogle Scholar
[14]Szeto, M., ‘Measure repleteness and mapping preservations’, J. Indian Math. Soc., to appear.Google Scholar
[15]Topsθe, F., Topology and measure, Springer Lacture Notes 133 (Springer-Verlag, Berlin, Heidelberg, New York, (1970)).CrossRefGoogle Scholar
[16]Varadarajan, V., ‘Measures on toplological spaces’, Amer. Math. Soc. Transl. Ser. 2. 48 (1965), 161–228.Google Scholar
[17]Wallman, H., ‘Lattices and topological spaces’, Ann. of Math. 42 (1938), 687–697.Google Scholar
[18]Yosida, K. and Hewitt, E., ‘Finitely additive measures’, Trans. Amer. Math. Soc. 72 (1952), 46–66.CrossRefGoogle Scholar
[19]Zaanen, A., An introduction to the theory of intergration, (North-Holland Publishing Co., Amsterdam, New York, (1958)).Google Scholar