The family of metric measure spaces can be endowed with the semigroup operation being the Cartesian product. The aim of this talk is to arrive at the generalisation of the fundamental theorem of arithmetics for metric measure spaces that provides a unique decomposition of a general space into prime factors. These results are complementary to several partial results available for metric spaces (like de Rham's theorem on decomposition of manifolds). Finally, the infinitely divisible and stable laws on the semigroup of metric measure spaces are characterised (joint work with S.N. Evans (Berkeley).