Mean values and differential equations

We show that the well-known equivalence between the mean-value theorem and harmonicity extends to arbitrary measures of compact support: a continuous function satisfies the generalized mean-value condition (1) with respect to a given measure if and only if it is annihilated by a certain system of homogeneous linear partial differential operators with constant coefficients determined by the measure. Extensions of this result are obtained, primarily in the direction of replacing systems of differential equations by a single equation.