The Topological Support of the z-Measures on the Thoma Simplex

The Thoma simplex $\Omega$ is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are probability measures on $\Omega$ depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit as it goes to 0, the z-measures turn into the Poisson–Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space $\Omega$.