We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of non-negative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study the analytic properties of this Halász–Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of non-negative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converge to the Halász–Székely barycenter of the corresponding distribution.
