Atomless support of a Pareto allocation

For an atomless measure of consumers with measurable, complete, transitive and continuous preference relations, every exactly feasible Pareto efficient consumption allocation admits nonzero quasi-equilibrium supporting prices. Countably many rational strict-improvement directions remove all exceptional null sets simultaneously. If their convex hull contained zero, sufficiently small disjoint cohorts could make those improvements in exactly balancing proportions, violating Pareto efficiency. A supporting hyperplane theorem then supplies the prices. Individual convexity of preferences is unnecessary.