= 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.
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