LetDbe an open subset of RNN≧ 2, such thatO∈Dand λ(D) < + ∞, where λ denotesN-dimensional Lebesgue measure. If the mean-value equalityholds for every integrable harmonic functionhonD. then accoeding to a theorem of KuranDis a ball of centreO. Here we show that the same conclusion holds if we assume the above mean-value equality only for positive integrable harmonic functions. Further, if the mean-value equality is assumed only for bounded harmonic functions, thenD=B=\E, whenBis an open ball andEis a polar set.