Simplicial Random Variables

We introduce a new geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the usual geometric realization through a natural map, which has probabilistic meaning : it associates to a random variable its probability mass function. This probability map’ function is proved to be a (Serre) fibration and a (weak) homotopy equivalence.