  In this article, we prove an extension of the mean value theorem and a
comparison theorem for subharmonic functions. These theorems are used to answer
the question whether we can conclude that two subharmonic functions which agree
almost everywhere on a surface with respect to the surface measure must
coincide everywhere on that surface. We prove that this question has a positive
answer in the case of hypersurfaces, and we also provide a counterexample in
the case of surfaces of higher co-dimension. We also apply these results to
Ahlfors-David sets and we prove other versions of the main results in terms of
measure densities.
