  In this paper, we study the eikonal equation in metric measure spaces, where
the inhomogeneous term is allowed to be discontinuous, unbounded and merely
$p$-integrable in the domain with a finite $p$. For continuous eikonal
equations, it is known that the notion of Monge solutions is equivalent to the
standard definition of viscosity solutions. Generalizing the notion of Monge
solutions in our setting, we establish uniqueness and existence results for the
associated Dirichlet boundary problem. The key in our approach is to adopt a
new metric, based on the optimal control interpretation, which integrates the
discontinuous term and converts the eikonal equation to a standard continuous
form. We also discuss the Holder continuity of the unique solution with respect
to the original metric under regularity assumptions on the space and the
inhomogeneous term.
