Let ? 6 -0 be either a power bounded radial measure with compact support on the unit disc D with ?(D) = 1 such that there is a - > 0 so that |ˆ(s)| 6 1 for every s 2 §(-) \ {0,1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {h 2 L 1 (D) | limn!1 h ⁄ ? n exists}. Let ?1,...,?n be measures on D with above mentioned properties. Here, we prove that if f 2 L1(Dn) satisfies an invariant volume mean value property with respect to ?1,...,?n, then f is n-harmonic.