Consider a action functional $S(g_{\mu \nu}, \psi)$ with spacetime metric $g_{\mu \nu}$ and fermionic matter fields $\psi$, which is invariant under diffeomorphisms/ coordinate transformations. However, in the partition function

$Z = \int D[g_{\mu \nu}] D[\psi] \mu e^{iS}$

the path integration measure $\mu$ is NOT diffeomorphism invariant and therefore, the covariant divergence of the energy-momentum tensor $T_{\mu \nu} = \frac{2 \delta S}{\sqrt{g} \delta g^{\mu \nu}}$ does not vanish.

Now I assume the following model: There are only distributions of the metric field allowed which correspond to triangulations of spacetime. That means the quantum gravitational theory should have an isomorphism of categories:

$c: Riem \rightarrow Tri$.

Here, $Riem$ is the category of Riemannian manifolds and $Tri$ the category of discrete manifolds that arise from triangulation of Riemannian manifolds. Hence, $\mu$ must be an indicator function that has the value 1 if a distribution of metric fields $g_{\mu \nu}(x)$ can be mapped functorially to category $Tri$ and has value 0 if this is not possible. Mathematically, the map $c$ must be a forgetful functor that forgets some kind of metric field distributions.

Question: Is that a plausible theory with diffeomorphism invariant action, but diffeomorphism non-invariant measure (a theory of gravitational anomaly)? Can on this way be constructed a quantum gravity theory (with category theoretic assumptions), e.g. causal triangulation theory?