Functional integration measure in quantum gravity

We study a peculiar regularization of quantum gravity at one-loop order intended to exhibit the properties of the functional measure. It reminds one of the Pauli-Villars technique in the sense that massive fields are introduced as regulators (and the mass is intended to go to infinity). The Pauli-Villars regulators are spin-2, -1, -½, and -0 particles coupled to gravity in a covariant way (mass terms included). We show that, under these conditions, the measure required in order to remove the maximal ultraviolet divergences [i.e., the divergences proportional to ${\delta }^{\mathrm{}\left(r\right)\mathrm{}}\mathrm{}\left(0\right)\mathrm{}$ if $r$ is the space-time dimension] is a product of measures of Fujikawa. Both the action and the measure of the functional integral are Becchi-Rouet-Stora (BRS) invariant. We consider also the regularization in the background-field formalism. We show that the measure of Fujikawa must be naturally generalized in order to be invariant under reparametrizations of the background as well as BRS invariant.