A note on logarithmic mean equicontinuity

We study the set of harmonic limits of empirical measures in topological dynamical systems. We obtain a characterization of unique ergodicity based of logarithmic (harmonic) mean convergence in place of Cesàro convergence. We introduce logarithmic mean equicontinuity and show that a topological dynamical system is logarithmically mean equicontinuous if and only if it is mean equicontinuous.