The fundamental theorem of arithmetic for metric measure spaces
Abstract: A metric measure space (mms) is simply a complete, separable metric
space equipped with a probability measure that has full support. A fundamental
insight of Gromov is that the space of such objects is much ‘tamer’ than the space of
complete, separable metric spaces per se because mms carry within themselves a
canonical family of approximations by finite structures: one takes the random mms
that arises from picking some number of points independently at random and
equipping it with the induced metric and uniform probability measure. A natural
(commutative and associative) binary operation on the space of mms is defined by
forming the Cartesian product of the two underlying sets equipped with the sum of
the two metrics and the product of the two probability measures. There is a
corresponding notion of a prime mms and an analogue of the fundamental theorem
of arithmetic in the sense that any mms has a factorization into countably many prime
mms which is unique up to the order of the factors. Moreover, a rich Fourier theory
enables one to analyze convolutions of probability measures on the space of mms
and obtain counterparts of classical results in the theory of infinitely divisible and
stable probability measures on Euclidean spaces due to Lévy, Itô, Hinčin, and
LePage. This is joint work with Ilya Molchanov (Bern).