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Generalization of a theorem of P. D. Finch's on integration of set-functions

Published online by Cambridge University Press:  09 April 2009

F. Cunningham Jr
F. Cunningham Jr
Affiliation:
Bryn Mawr College Bryn Mawr, Pennsylvania
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Let Mbe a σ-field of subsets of a space X. A partition of Xmeans a countable partition Π of Xinto sets belonging to M; the set of partitions is directed by refinement. A. Kolmogoroff in 1930 [1] discussed an integral (Moore-Smith limit as Π gets finer) for set-functions F defined on M. When it exists, IF is σ-additive, and if by chance F is already σ-additive, then IF = F

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 4 , November 1966 , pp. 460 - 465
Copyright
Copyright © Australian Mathematical Society 1966

References

[1]Kolmogoroff, A., ‘Untersuchung uber das Integralbegriff’, Math. Annalen 103 (1930), 654–698.CrossRefGoogle Scholar
[2]Finch, P. D., ‘Integration of real-valued set functions in abstract spaces’, J. Australian Math. Soc. 4 (1964), 202–213.CrossRefGoogle Scholar
[3]Finch, P. D., ‘A generalization of the Radon-Nikodym theorem’, J. Australian Math. Soc. 5 (1965), 17–24.CrossRefGoogle Scholar