Kakutani's theorem (measure theory)

In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable product measures. It gives an if and only if characterisation of when two such measures are equivalent, and hence it is extremely useful when trying to establish change-of-measure formulae for measures on function spaces. The result is due to the Japanese mathematician Shizuo Kakutani. Kakutani's theorem can be used, for example, to determine whether a translate of a Gaussian measure is equivalent to (only when the translation vector lies in the Cameron–Martin space of ), or whether a dilation of is equivalent to (only when the absolute value of the dilation factor is 1, which is part of the Feldman–Hájek theorem).

Statement of the theorem

For each , let and be measures on the real line , and let and be the corresponding product measures on . Suppose also that, for each , and are equivalent (i.e. have the same null sets). Then either and are equivalent, or else they are mutually singular. Furthermore, equivalence holds precisely when the infinite product

has a nonzero limit; or, equivalently, when the infinite series

converges.

References

  • Bogachev, Vladimir (1998). Gaussian Measures. Mathematical Surveys and Monographs. 62. Providence, RI: American Mathematical Society. doi:10.1090/surv/062. ISBN 0-8218-1054-5. (See Theorem 2.12.7)
  • Kakutani, Shizuo (1948). "On equivalence of infinite product measures". Ann. Math. 49: 214–224. doi:10.2307/1969123.
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