  We adopt the standard definition of diffeomorphism for Regge gravity in D=2
and give an exact expression of the Liouville action in the discretized case.
We also give the exact form of the integration measure for the conformal
factor. In D>2 we extend the approach to any family of geometries described by
a finite number of parameters. The ensuing measure is a geometric invariant and
it is also invariant in form under an arbitrary change of parameters.
