  We introduce a notion of barycenter of a probability measure related to the
symmetric mean of a collection of nonnegative real numbers. Our definition is
inspired by the work of Hal\'asz and Sz\'ekely, who in 1976 proved a law of
large numbers for symmetric means. We study analytic properties of this
Hal\'asz-Sz\'ekely barycenter. We establish fundamental inequalities that
relate the symmetric mean of a list of nonnegative 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 converges to the Hal\'asz-Sz\'ekely
barycenter of the corresponding distribution.
