  We study the obstacle problem for unbounded sets in a proper metric measure
space supporting a (p,p)-Poincare inequality. We prove that there exists a
unique solution. We also prove that if the measure is doubling and the obstacle
is continuous, then the solution is continuous, and moreover p-harmonic in the
set where it does not touch the obstacle. This includes, as a special case, the
solution of the Dirichlet problem for p-harmonic functions with Sobolev type
boundary data.
