The space of homogeneous probability measures on ${\overline{\mathrm{\Gamma }\mathrm{\backslash }X}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^S$ is compact

In this paper we prove that the space of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space is compact. As an application, we explain some consequences for the distribution of weakly special subvarieties of Shimura varieties.