Generalization of the Change of Variables Formula with Applications to Residual Flows

The paper’s Theorem 4 states and (essentially) proves the CVF for L-diffeomorphisms, yielding pf(x) = pZ(f^{-1}(x)) |det J f^{-1}(x)| almost everywhere, by removing null sets so the restriction is a diffeomorphism and then applying the standard CVF; see the definition of L-diffeomorphism and Theorem 4 with proof in the main text and Appendix A (e.g., Definition 2 and Theorem 4; proof lines leading to eqs. (8)–(11)) . The candidate solution proves the same result by a patchwise change-of-variables argument on countably many local inverse charts and careful disjointization, avoiding any set-theoretic slippage. The only issue we found in the paper is the asserted set equality f^{-1}(B) \ NZ = f^{-1}(B \ NX) (eq. (9)), which need not hold absent an assumption like f(NZ) ⊆ NX; however, this can be fixed with a subset relation plus a measure-zero correction, leaving the main theorem intact . Hence both are correct; the model’s proof is different (more granular) and fully resolves the technical point.