Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T01:14:21.356Z Has data issue: false hasContentIssue false
  • English
  • Français

Decompositions of Submeasures

Published online by Cambridge University Press:  20 November 2018

Cecilia H. Brook
Cecilia H. Brook*
Affiliation:
Northern Illinois University, Dekalb, Illinois
Rights & Permissions [Opens in a new window]

Extract

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 [4] we showed that one can tell whether a submeasure on a Boolean algebra has a control measure or is pathological by comparing the Fréchet-Nikodym topology it generates to the universal measure topology of Graves. We then wondered if a submeasure could be decomposed into a part with a control measure and a part which is pathological or zero. This led to the problem of finding a Lebesgue decomposition for a submeasure on an algebra of sets with respect to a Fréchet-Nikodym topology.

In [6] Drewnowski proved a Lebesgue decomposition theorem for exhaustive submeasures with respect to “additivities” and a similar theorem for exhaustive Fréchet-Nikodym topologies. He asked if an exhaustive Fréchet-Nikodym topology could be decomposed with respect to another Fréchet-Nikodym topology. In [12] Traynor showed that the answer is “yes”.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 4 , 01 August 1984 , pp. 577 - 590
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Aleksyuk, V. N., Theorem on the minorant, Math. Notes Acad. Sci. USSR 21 (1977), 336–340.Google Scholar
2. Brook, C. H., On the universal measure space, J. Math. Anal. Appl. 85 (1982), 584–598.Google Scholar
3. Brook, C. H., Continuity properties of vector measures, J. Math. Anal. Appl. 86 (1982), 268–280.Google Scholar
4. Brook, C. H., The control measure problem and the universal measure topology, Rocky Mt. J. Math 13 (1983), 265–272.Google Scholar
5. Drewnowski, L., Topological rings of sets, continuous set functions, integration I, II, III, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20 (1972), 269–276, 277–286, 439–445.Google Scholar
6. Drewnowski, L., Decompositions of set functions, Studia Math. 48 (1973), 23–48.Google Scholar
7. Drewnowski, L., On control submeasures and measures, Studia Math. 50 (1974), 203–224.Google Scholar
8. Fremlin, D. H., The control measure problem, (note of March, 1978), University of Essex, Colchester, England.Google Scholar
9. Graves, W. H., On the theory of vector measures, Amer. Math. Soc. Memoirs 195 (1977).Google Scholar
10. Maharam, D., An algebraic characterization of measure algebras, Ann. Math. 48 (1947), 154–167.Google Scholar
11. Traynor, T., S-bounded additive set functions, in Vector and operator-valued measures and applications (Proc. Sympos., Snowbird Resort, Alta, Utah, 1972) (Academic Press, New York), 355–365.Google Scholar
12. Traynor, T., The Lebesgue decomposition for group-valued set functions, Trans. Amer. Math. Soc 220 (1976), 307–319.Google Scholar
13. Wilansky, A., Topology for analysis (Ginn, Waltham, Mass., 1970).Google Scholar