Reweighting metric measure spaces and Onsager-Machlup

Given a metric measure space $M:=(X,d,\mu )$ the Onsager-Machlup (OM) functional is a real valued function that has been seen as a generalized notion of a probability density function. The effect of reweighting the measure on OM functionals has been studied, however analogous reweightings of the metric to the best of our knowledge remain open. In this short note, we prove a transformation formula for OM functionals on geodesic metric measure spaces under reweighting of both the metric and the measure.