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 "
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
" 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
Japanese
Japanese may refer to:
* Something from or related to Japan, an island country in East Asia
* Japanese language, spoken mainly in Japan
* Japanese people, the ethnic group that identifies with Japan through ancestry or culture
** Japanese diaspor ...
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 ...
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 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
, 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
* (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