Rigorous definition of the gravitational path integral measure

Develop a mathematically rigorous definition of the path-integral measure over the space of metrics for the Euclidean gravitational path integral, so that integration over geometries is well defined beyond the semiclassical saddle-point approximation and does not rely on ad hoc prescriptions.

Background

The notes formulate the gravitational path integral formally as an integral over metrics and matter fields weighted by the Einstein–Hilbert action (plus boundary terms and matter), and then immediately move to the semiclassical approximation because the measure over metrics is not defined. This prevents a full, non-perturbative construction of the path integral and limits the framework to saddle-point evaluations and perturbative expansions around classical solutions.

A rigorous measure would enable controlled inclusion of quantum fluctuations and potentially resolve ambiguities tied to gauge redundancies and contour choices in complexified metric space.