  We consider the projectivization of Minkowski space with the analytic
continuation of the hyperbolic metric and call this an extended hyperbolic
space. We can measure the volume of a domain lying across the boundary of the
hyperbolic space using an analytic continuation argument. In this paper we show
this method can be further generalized to find the volume of a domain with
smooth boundary with suitable regularity in dimension 2 and 3. We also discuss
that this volume is invariant under the group of hyperbolic isometries and that
this regularity condition is sharp.
