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 , or whether a dilation of is equivalent to .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. 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.
