Mass localization

Abstract

For a given class $\mathcal{F}$ of closed sets of a measured metric space $(E,d,\mu)$, we want to find the smallest element $B$ of the class $\mathcal{F}$ such that $\mu(B)\geq 1-\alpha$, for a given $0<\alpha<1$. This set $B$ \textit{localizes the mass} of $\mu$. Replacing the measure $\mu$ by the empirical measure $\mu_n$ gives an empirical smallest set $B_n$. The article introduces a formal definition of small sets (and their size) and study the convergence of the sets $B_n$ to $B$ and of their size.