Ideal Convergence of Regular Uncertainty Distributions in Lᵖ-Distance

Let 1≤ p<∞. On the class of regular uncertainty distributions with p-integrable inverse distributions, we consider the Lᵖ-distance between the inverse distributions. Using the standard quantile representation, we identify the completion of this metric space with the closed cone of almost-everywhere equivalence classes of nondecreasing functions in Lᵖ(0,1) and characterize its relatively compact subsets by uniform p-integrability. For an admissible ideal I on N, the Cauchy criterion for complete metric spaces gives the equivalence of ideal convergence and the ideal Cauchy condition in the completed space. The classical additive-property criterion also applies: ideal convergence is realized along a subsequence indexed by some M ∈ F(I) for every sequence if and only if I has the additive property. Under uniform p-integrability, convergence in dₚ is equivalent to pointwise ideal convergence of the associated distribution functions at every continuity point. An endpoint-concentration example shows that pointwise distributional convergence alone does not imply convergence in dₚ. Finally, expected values satisfy a sharp Lipschitz estimate, and d₂ reduces to the Euclidean distance between the location and scale parameters of normal uncertainty distributions.