Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-30T20:42:04.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Openness of vector measures and their integral maps

Part of: Maps and general types of spaces defined by maps Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  09 April 2009

Andrzej Spakowski
Andrzej Spakowski
Affiliation:
Institute of Mathematics Pedagogical University, Oleska 48, 45-951 Opole, 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.

We prove that finite dimensional nonatomic vector measures and their integral maps are open maps. These results can be found in the literature, but unfortunately the proofs presented there are not complete.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 351 - 359
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Anantharaman, R., ‘On exposed points of the range of a vector measure’, Proc. Snowbird sympos. on vector and operator valued measures and applications, pp. 7–22 (Academic Press, 1973).CrossRefGoogle Scholar
[2]Anantharaman, R. and Garg, K. M., ‘Some topological properties of vector measures and their integral maps’, J. Austral. Math. Soc. Ser. A 23 (1977), 453–466.CrossRefGoogle Scholar
[3]Anantharaman, R. and Garg, K. M., ‘On the range of a vector measure’, Bull. Math. Soc. Sci. Math. Roumainie 22 (1978), 115–132.Google Scholar
[4]Bartle, R. G., Dunford, N. and Schwartz, J. T., ‘Weak compactness and vector measures’, Canad. J. Math. 7 (1955), 289–305.Google Scholar
[5]Bolker, E. D., ‘A class of convex bodies’, Trans. Amer. Math. Soc. 145 (1969), 323–345.CrossRefGoogle Scholar
[6]Bula, W., ‘On metrizability of continuous images of compact ordered spaces’, Fund. Math. 123 (1984), 21–27.CrossRefGoogle Scholar
[7]Engelking, R., General topology (PWN, Warszawa, 1977).Google Scholar
[8]Glicksberg, P., ‘A controlled approach to Liapounoff convexity’, J. Math. Anal. Appl. 94 (1983), 193–196.Google Scholar
[9]Hájek, O., ‘Notes on quotient maps’, Comment. Math. Univ. Carolinae 7 (1966), 319–323.Google Scholar
[10]Halmos, P. R., ‘The range of a vector measure’, Bull. Amer. Math. Soc. 54 (1948), 416–421.CrossRefGoogle Scholar
[11]Jerison, M., ‘A property of extreme points of compact convex sets’, Proc. Amer. Math. Soc. 5 (1954), 782–783.CrossRefGoogle Scholar
[12]Karafiat, A., ‘On the continuity of a mapping inverse to a vector measure’, Comment. Math. Prace Mat. 18 (1974), 37–43.Google Scholar
[13]Kluvánek, I. and Knowles, G., Vector measures and control systems (North-Holland, New York, 1976).Google Scholar
[14]Kuratowski, K., Topology I (Academic Press, New York, 1966).Google Scholar
[15]Liapunov, A. A., ‘Sur les fonctions vecteur completement additives’ (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 465–478.Google Scholar
[16]Samet, D., ‘Vector measures are open maps’, Math. Oper. Res. 9 (1984), 471–474.Google Scholar
[17]Sikorski, R., ‘Closure homomorphisms and interior mappings’, Fund. Math. 41 (1954), 12–20.Google Scholar
[18]Armstrong, T. E., ‘Openness of finitely additive vector measures as mappings’, preprint.Google Scholar