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A metric measure space (mms) is just 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 a mms carries within itself a natural family of approximations by finite structures: one takes the random mmss that arise by picking some number of points independently at random and equipping them with the induced metric and uniform probability measure.  A natural (commutative and associative) binary operation on the space of mmss 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 mmss 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 Levy, Ito, Hincin, and LePage.  The talk is based on a joint paper with Ilya Molchanov (Bern) that will appear in Trans. Amer. Math. Soc.