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A Class of Singular Functions

Published online by Cambridge University Press:  20 November 2018

Harold S. Shapiro
Harold S. Shapiro*
Affiliation:
The University of Michigan, Ann Arbor, Michigan
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Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1425 - 1431
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This paper was written in the Soviet Union while the author participated in the exchange program with the Soviet Academy of Sciences. He is indebted to the National Academy of Sciences for financial support.

References

1. Halmos, P., Measure theory (Van Nostrand, Princeton, N.J., 1950).10.1007/978-1-4684-9440-2CrossRefGoogle Scholar
2. Kakutani, S., Oequivalence of infinite product measures, Ann. of Math. (2) 1+9 (1948), 214–226 Google Scholar