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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

Information

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

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