Ideal Convergence of Regular Uncertainty Distributions in $L^p$-Distance

Let $1\leq p<\infty$. On the class of regular uncertainty distributions with $p$-integrable inverse distributions, we consider the $L^p$-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 $L^p(0,1)$ 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 $M \in\mathcal F(I)$ for every sequence if and only if $I$ has the additive property. Under uniform $p$-integrability, convergence in $d_p$ 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 $d_p$. Finally, expected values satisfy a sharp Lipschitz estimate, and $d_2$ reduces to the Euclidean distance between the location and scale parameters of normal uncertainty distributions.