Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, then
A converse of this result is due to Kuran : if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such that
for every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein , theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.