This paper explores a Dirichlet type problem on metric
measure spaces. The problem is to find a Sobolev-type function
that minimizes the energy integral within a class of
"Sobolev" functions that agree with the boundary function
outside the domain of the problem. This is the analogue of the
Euler-Lagrange formulation in the classical Dirichlet
problem. It is shown that, under certain geometric constraints
on the measure imposed on the metric space, such a solution
exists. Under the condition that the space has many
rectifiable curves, the solution is unique and satisfies the
weak maximum principle.