An atom obstruction to stochastic non-diversification

In models with infinite mean, weighted averages of independent risks can stochastically dominate the underlying marginal distribution. This paper studies the class D⁻ of probability measures on the real line for which this non-diversification phenomenon holds across all finite convex combinations. We show that any law in D⁻ satisfies μ({a}) · μ((-∞, a)) = 0 for every a ∈ ℝ, so that no probability mass can lie strictly below an atom. This gives a negative answer to the discrete-law question posed by Müller (2025), proving that the only purely discrete distributions in D⁻ are Dirac measures. The conclusion follows from a two-copy comparison along positive weights tending to zero, with no moment or support-bound assumptions. Lean 4 formalization and replication package: https://github.com/Theophilus1030/lean-nondiversification