On Furstenberg's characterization of harmonic functions on symmetric spaces

We provide a necessary and sufficient condition on a radial probability measure μ on a symmetric space for which f = f * μ, f bounded, implies that f is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that if f is a bounded function on a symmetric space which satisfies f = f * μ for some radial absolutely continuous probability measure μ, then f is harmonic.