Volume densities with the mean value property for harmonic functions

On a bounded domain U in ${\mathbb{R}^d}$ containing the origin, probability measures $\mu$ which have a density w with respect to Lebesgue measure and satisfy $h(0) = \smallint h\;d\mu$ for every bounded harmonic function on U are studied. A domain U is constructed such that $\inf w(U) = 0$ for any such measure. (This solves a problem proposed by A. Cornea.) If, however, U has smooth boundary, then $\mu$ having a density $w \in {\mathcal{C}^\infty }(U)$ which is bounded away from zero on U can be constructed. On the other hand, for arbitrary U it is always possible to choose a strictly positive $w \in {\mathcal{C}^\infty }(U)$ tending to zero at $\partial U$.