Let ${ au};{ eq};delta_0$ be either a power bounded radial measure with compact support on the unit disc D with $ au(D);=;1$ such that there is a $delta$ > 0 so that ${mid}hat{ au}(s){mid};{ eq};1$ for every $s;{in};Sigma(delta)$ {0,1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {$h;{in};L^{infty}(D);{mid};lim_{n{ ightarrow}{infty}};h;*; au^n$ exists}. Let $ au_1$, ..., $ au_n$ be measures on D with above mentioned properties. Here, we prove that if $f;{in};L^{infty}(D^n)$ satisfies an invariant volume mean value property with respect to $ au_1$, ..., $ au_n$, then f is n-harmonic.