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

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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

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  • Bogachev, Vladimir (1998). Gaussian Measures. Mathematical Surveys and Monographs. Vol. 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.