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In measure theory, a branch of mathematics, Kakutani's theorem is a fundamental result on the equivalence or mutual singularity of countable
product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of ...
s. It gives an "
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" 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 space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
s. The result is due to the
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mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Shizuo Kakutani was a Japanese-American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institute for ...
. Kakutani's theorem can be used, for example, to determine whether a translate of a
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named ...
\mu is equivalent to \mu (only when the translation vector lies in the Cameron–Martin space of \mu), or whether a dilation of \mu is equivalent to \mu (only when the absolute value of the dilation factor is 1, which is part of the
Feldman–Hájek theorem In probability theory, the Feldman–Hájek theorem or Feldman–Hájek dichotomy is a fundamental result in the theory of Gaussian measures. It states that two Gaussian measures \mu and \nu on a locally convex space X are either equivalent meas ...
).


Statement of the theorem

For each n \in \mathbb, let \mu_ and \nu_ be measures on the real line \mathbb, and let \mu = \bigotimes_ \mu_n and \nu = \bigotimes_ \nu_n be the corresponding product measures on \mathbb^\infty. Suppose also that, for each n \in \mathbb, \mu_n and \nu_n are equivalent (i.e. have the same null sets). Then either \mu and \nu are equivalent, or else they are mutually singular. Furthermore, equivalence holds precisely when the infinite product :\prod_ \int_ \sqrt \, \mathrm \nu_n has a nonzero limit; or, equivalently, when the infinite series :\sum_ \log \int_ \sqrt \, \mathrm \nu_n converges.


References

* (See Theorem 2.12.7) * {{cite journal , last = Kakutani , first = Shizuo , title = On equivalence of infinite product measures , journal = Ann. Math. , volume = 49 , pages = 214–224 , year = 1948 , doi=10.2307/1969123 Probability theorems Theorems in measure theory