Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T15:27:06.261Z Has data issue: false hasContentIssue false
  • English
  • Français

Some topological properties of vector measures and their integral maps

Published online by Cambridge University Press:  09 April 2009

R. Anantharaman  and
K. M. Garg
R. Anantharaman
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada.
K. M. Garg
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada.
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.

In the present paper we prove that every finite dimensional non-atomic measure ν is open and monotone (viz. ν–1 preserves connected sets) relative to the usual Fréchet-Nikodým topology on its domain and the relative topology on its range. An arbitrary finite dimensional measure is found on the other hand to be biquotient.

Given a vector measure ν, we further investigate the properties of its integral map Tν: φ → ∫ φdν defined on the set of functions φ in L1(|ν|) for which φ(s) ∈ [0,1] |ν|-almost everywhere. When ν is finite dimensional, Tν is found to be always open. In general, when Tν is open, the set of extreme points of the closed convex hull of the range of ν is proved to be closed, and when ν and Tν are both open, the range of ν in itself is closed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 4 , June 1977 , pp. 453 - 466
Copyright
Copyright © Australian Mathematical Society 1977

References

Anantharaman, R. (1973), ‘On exposed points of the range of a vector measure’, Snowbird symposium on vectro and operator valued measures and application (Academic Press, New York).Google Scholar
Anantharaman, R. (1974), On the range of a vector measure (Thesis, Univ. of Alberta, Edmonton).Google Scholar
Anantharaman, R. and Garg, K. M., ‘On the range of a vector measure’, (submitted).Google Scholar
Bartle, R. G., Dunford, N. and Schwartz, J. T. (1955), ‘Weak compactness and vector measures’, Canad. J. Math. 7, 289–305.CrossRefGoogle Scholar
Birkhoff, G. (1967), Lattice Theory (A.M.S., Providence, R.I., 3rd ed. 1967).Google Scholar
Garg, K. M. (1977), ‘Properties of connected functions in terms of their levels’, Fund. Math. 97 (to appear).CrossRefGoogle Scholar
Hájek, O. (1967), ‘Notes on quotient maps’, Comm. Math. Univ. Carolinae 7, 319–323.Google Scholar
Halmos, P. R. (1947), ‘On the set of values of a finite measure’, Bull. Amer. Math. Soc. 53, 138–141.CrossRefGoogle Scholar
Halmos, P. R. (1948), ‘The range of a vector measure’, Bull. Amer. Math. Soc. 54, 416–421.CrossRefGoogle Scholar
Halmos, P. R. (1950), Measure Theory (Van Nostrand, New York, 1950).CrossRefGoogle Scholar
Hoffman-Jørgensen, J. (1971), ‘Vector measures’, Math. Scand. 28, 5–32.CrossRefGoogle Scholar
Jerison, M. (1954), ‘A property of extreme points of compact convex sets’, Proc. Amer. Math. Soc. 5, 782–783.CrossRefGoogle Scholar
Karafiat, A. (1974), ‘On the continuity of a mapping inverse to a vector measure’, Prace Mat. 18, 37–43.Google Scholar
Kingman, J. F. C. and Robertson, A. P. (1968), ‘On a theorem of Lyapounov’, J. Lond. Math. Soc. 43, 347–351.CrossRefGoogle Scholar
Kluvánek, I. and Knowles, G. (1974), Vector measures and control systems (Flinders Univ. of South Australia, Bedford Park).Google Scholar
Knowles, G. (1973), Vector Measures and bang-bang control, Thesis, Flinders Univ. of South Australia, Bedford Park.Google Scholar
Kuratowski, K. (1966), Topology I (Academic Press, New York, Eng. Tr., 1966).Google Scholar
Kuratowski, K. (1968), Topology II (Academic Press, New York, Eng. Tr., 1968).Google Scholar
Liapounoff, A. A. (1940), ‘Sur les fonctions-vecteurs complètement additives’, Izv. Akad. Nauk S.S.S.R. (Serie Mat.) 4, 465–478.Google Scholar
Lindenstrauss, J. (1966), ‘A short proof of Liapounoff's convexity theorem’, J. Math. Mech. 15 971–972.Google Scholar
Michael, E. (1951), ‘Topologies on spaces of subsets’, Trans. Amer. Math. Soc. 71, 152–182.CrossRefGoogle Scholar
Michael, E. (1959), ‘A theorem on semi-continuous set-valued functions’, Duke Math. J. 26, 647–651.CrossRefGoogle Scholar
Michael, E. (1968), ‘Bi-quotient maps and cartesian products of quotient maps’, Ann. Inst. Fourier. Grenoble 18, 287–302.CrossRefGoogle Scholar
Schmets, J. (1966), ‘Sur une generalisation d'une théorème de Lyapounov’, Bull. Soc. Roy. Sci Liege, 35eannee, 185–194.Google Scholar
Sikorski, R. (1955), ‘Closure homomorphisms and interior mappings’, Fund. Math. 41, 12–20.CrossRefGoogle Scholar
Whyburn, G. T. (1967), Analytic Topology (A.M.S., Providence, R.I., 2nd ed. 1967).Google Scholar
Whyburn, G. T. (1970), ‘Dynamic topology’, Amer. Math. Monthly 77, 556–570.CrossRefGoogle Scholar