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Charges on Boolean algebras and almost discrete spaces

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

M. Bhaskara Rao  and
K. P. S. Bhaskara Rao
M. Bhaskara Rao
Affiliation:
Department of Probability & Statistics, The University, Sheffield, S3 7RH.
K. P. S. Bhaskara Rao
Affiliation:
Indian Statistical Institute, 203 B. T. Road, Calcutta 35.
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Extract

The notion of nonatomicity of a measure on a Boolean σ-algebra is an important concept in measure theory. What could be an appropriate analogue of this notion for charges defined on Boolean algebras is one of the topics dealt with in this paper. Analogous to the decomposition of a measure on a Boolean σ-algebra into atomic and nonatomic parts, no decomposition of charges is available in the literature. We provide a simple proof of such a decomposition. Next, we study the conditions under which a Boolean algebra admits certain types of charges. These conditions lead us to give a characterisation of superatomic Boolean algebras. Babiker' [1] almost discrete spaces are connected with superatomic Boolean algebras and a generalisation of one of his theorems is obtained. A counterexample is also provided to disprove one of his theorems. Finally, denseness problems of certain types of charges are studied.

MSC classification

Secondary: 28A60: Measures on Boolean rings, measure algebras
Type
Research Article
Information
Mathematika , Volume 20 , Issue 2 , December 1973 , pp. 214 - 223
Copyright
Copyright © University College London 1973

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References

1. Babiker, A. G. A., “On almost discrete spaces”, Mathematika, 18 (1971), 163–167.CrossRefGoogle Scholar
2. Rao, K. P. S. Bhaskara and Rao, M. Bhaskara, “Existence of nonatomic charges”, to appear in J. Australian Math. Soc.Google Scholar
3. Rao, M. Bhaskara and Rao, K. P. S. Bhaskara, “Borel σ-algebra on [0, Ω]”, Manuscripta Mathematica, 5 (1971), 195–198.Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, 1960).CrossRefGoogle Scholar
5. Halmos, P. R., Measure theory (Van Nostrand, 1955).Google Scholar
6. Halmos, P. R., Lectures on Boolean algebras (Van Nostrand, 1967).Google Scholar
7. Kelley, J. L., General topology (Van Nostrand, 1955).Google Scholar
8. Knowles, J. D., “On the existence of nonatomic measures”, Mathematika, 14 (1967), 62–67.CrossRefGoogle Scholar
9. Rudin, M. E., “A subset of the countable ordinals”, Amer. Math. Monthly, 64 (1957), 351.CrossRefGoogle Scholar
10. Rudin, W., “Continuous functions on compact spaces without perfect subsets”, Proc. Amer. Math. Soc., 8 (1957), 39–42.CrossRefGoogle Scholar
11. Sikorski, R., Boolean algebras (Springer-Verlag, 1969).CrossRefGoogle Scholar
12. Sobczyk, A. and Hammer, P. C., “A decomposition of additive set functions”, Duke Math. J., 11 (1944), 839–846.Google Scholar
13. Semadeni, Z., Banach spaces of continuous functions, Volume 1 (Polish Sci. Publishers, 1971).Google Scholar