  We review the construction of the path integral and the corresponding
effective action for the Regge formulation of General Relativity under the
assumption that the short-distance structure of the spacetime is not a smooth
4-manifold, but a picewise linear manifold based on a triangulation of a smooth
4-manifold. We point out that the exponentially damped 4-volume path-integral
measure does not give a finite path integral, although it can be used for the
construction of the perturbative effective action. We modify the 4-volume
measure by multiplying it by an inverse power of the product of the
edge-lengths such that the new measure gives a finite path integral while it
retains all the nice features of the unmodified measure.
