What is typical?

Abstract

Let $\xi$ be a random measure on a locally compact second countable topological group and let $X$ be a random element in a measurable space on which the group acts. In the compact case, we give a natural definition of the concept that the origin is a typical location for $X$ in the mass of $\xi$, and prove that when this holds the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.