Given a Polish space X , a countable Borel equivalence relation E on X , and a Borel cocycle ρ : E → (0,∞), we characterize the circumstances under which there is a probability measure μ on X such that ρ(φ−1(x), x) = [d(φ∗μ)/dμ](x) μ-almost everywhere, for every Borel injection φ whose graph is contained in E.