Direct sums and decompositions of Gromov's pyramids

Gromov introduced the notion of a pyramid as a generalization of a metric measure space, based on the idea of the concentration of measure phenomenon. In this paper, we introduce the concept of a direct sum of pyramids, which naturally appears as a limit of a sequence of metric measure spaces whose measures concentrate on finitely or countably many regions, with the distances between these regions diverging to infinity. As one of our main results, we prove that any pyramid admits a unique direct sum decomposition. Moreover, as an application, we establish the method for checking whether a given pyramid is an extended metric measure space.