A Note on Restricted Selection Set from Random Interval (2512.04539v1)

Abstract: We study restricted selection sets of random intervals in $\R^1$ defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we define the restricted selection set and establish its existence, basic structure and influence on bounding moments and quantiles. In particular, we give conditions under which any mean (or quantile) in the Aumann expectation range can be attained by a measurable selection. We characterize the induced ranges of means, medians, and event probabilities. The analysis is carried out in a minimal one-dimensional random-set framework inspired by the classical theory of Aumann integrals. We also outline extensions to higher-order moment and general quantile restrictions.