Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1; 1)-Poincare inequality. The notion of harmonicity is based on the Dirichlet form dened in terms of a Cheeger dierentiable structure. By studying ne properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss {Green type formula for sets of nite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss {Green formula we introduce a suitable notion of the interior normal trace of a regular ball.