Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-16T05:20:14.807Z Has data issue: false hasContentIssue false
  • English
  • Français

When is the algebra of regular sets for a finitely additive borel measure a α-algebra?

Part of: Peculiar spaces Classical measure theory Set functions and measures on spaces with additional structure

Published online by Cambridge University Press:  09 April 2009

Thomas E. Armstrong
Thomas E. Armstrong
Affiliation:
Department of Mathematical Sciences Northern Illinois UniversityDe Kalb, Illinois 60115, 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.

It is shown that hte algebra of regular sets for a finitely additive Borel measure μ on a compact Hausdroff space is a σ-algebra only if it includes the Baire algebra and μ is countably additive onthe σ-algebra of regular sets. Any infinite compact Hausdroff space admits a finitely additive Borel measure whose algebra of regular sets is not a σ-algebra. Although a finitely additive measure with a σ-algebra of regular sets is countably additive on the Baire σ-algebra there are examples of finitely additive extensions of countably additive Baire measures whose regular algebra is not a σ-algebra. We examine the particular case of extensions of Dirac measures. In this context it is shown that all extensions of a {0, 1}-valued countably additive measure from a σ-algebra to a larger σ-algebra are countably additive if and only if the convex set of these extensions is a finite dimensional simplex.

Keywords

Borel measureregularityextensions of measurescompletion regular compact spaceBorel regular compact space.

MSC classification

Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.) 28A60: Measures on Boolean rings, measure algebras 54G10: $P$-spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 3 , December 1982 , pp. 374 - 385
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Armstrong, T. E. and Prikry, K., ‘Residual measures’, Illinois J. Math. 22 (1978), 64–78.CrossRefGoogle Scholar
[2]Armstrong, T. E. and Prikry, K., ‘κ-finiteness and κ-additivity of measures on sets and left-invariant measures on discrete groups’, Proc. Amer. Math. Soc. 80 (1980), 105–112.Google Scholar
[3]Armstrong, T. E. and Prikry, K., ‘Liapounoff's theorem for non-atomic bounded, finitely additive, finite dimensional, vector valued measures’, Trans. Amer. Math. Soc. 266 (1981). 499–514.Google Scholar
[4]Babiker, A. G. A. G. and Knowles, J. D., ‘An example concerning completion regular measures. images of measurable sets and measurable selections’, Mathematika 25 (1978), 120–124.CrossRefGoogle Scholar
[5]Berberian, S. K., Measure and integration, (Chelsea, New York, 1970).Google Scholar
[6]Billingsley, P., Convergence of probability measures, (Wiley, New York, 1968).Google Scholar
[7]Dashiell, F., Hager, A. and Henriksen, M., ‘Order-Cauchy completions of rings and vector lattices of continuous functions’, Canad. J. Math. 32 (1980), 657–685.CrossRefGoogle Scholar
[8]Gardner, R. J., ‘The regularity of Borel measures and Borel measure compactness’, Proc. London Math. Soc. 30 (1975), 95–113.CrossRefGoogle Scholar
[9]Gillman, L. and Jerison, M., Rings of continuous functions, (Van Nostrand, Princeton, N. J., 1960).CrossRefGoogle Scholar
[10]Graves, W. and Wheeler, R. F., ‘On the Grothendieck and Nikodym properties for algebras of Baire, Borel and universally measurable sets’, preprint.Google Scholar
[11]Halmos, P., Measure theory, (Van Nostrand, Princeton, N. J., 1950).CrossRefGoogle Scholar
[12]Halmos, P., Lectures on Boolean algebras, (Van Nostrand, Princeton, N. J., 1963).Google Scholar
[13]Hewitt, E. and Yosida, K., ‘Finitely additive measures’, Trans. Amer. Math. Soc. 72 (1952). 46–66.Google Scholar
[14]Jacobs, K., Measure and integral, (Academic Press, New York, 1978).Google Scholar
[15]Kelley, J. L., General topology, (Van Nostrand, Princeton, N. J., 1955).Google Scholar
[16]Kupka, J., ‘Uniform boundedness principles for regular Borel vector measures’, J. Austral. Math. Soc. 29 (1980), 206–218.CrossRefGoogle Scholar
[17]Levy, R., ‘Almost P-spaces’, Canad. J. Math. 29 (1977), 284–288.CrossRefGoogle Scholar
[18]Martin, A. F., ‘A note on monogenic Baire measures’, Amer. Math. Monthly 84 (1977), 554–555.CrossRefGoogle Scholar
[19]Okada, S. and Okasaki, Y., ‘On measure-compactness and Borel measure-compactness’, Osaka J. Math. 15 (1978), 183–191.Google Scholar
[20]Plachky, D., ‘Extremal and monogenic additive set functions’, Proc. Amer. Math. Soc. 54 (1976), 193–196.CrossRefGoogle Scholar
[21]Schachermayer, W., ‘Eberlein-compacts et espaces de Radon’, Bol. Ann. Sci. Univ. Clermont 61 (1976), 129–145.Google Scholar
[22]Schachermayer, W., ‘On compact spaces which are not c-spaces’, Bol. Soc. Mat. Mexicana 22 (1977), 60–63.Google Scholar
[23]Semadeni, Z., Banach spaces of continuous functions, (Polish Scientific Press, Warsaw, 1971).Google Scholar
[24]Sikorski, R., Boolean algebras, (Springer, New York, 1969).CrossRefGoogle Scholar
[25]Topsoe, F., ‘Approximating pavings and construction of measures’, Colloq. Math. 42 1979, 377–385.CrossRefGoogle Scholar
[26]Traynor, T., ‘The Lebesgue decomposition for group valued set functions’, Trans. Amer. Math. Soc. 220 (1976), 307–319.CrossRefGoogle Scholar
[27]Veksler, A. E., ‘Functional characteristics of P'-spaces’, Comment. Math. Univ. Carolinae 18 (1977), 363–366.Google Scholar
[28]Walker, R. C., The Stone-Çech compactification, (Springer, New York, 1974).CrossRefGoogle Scholar