Weakly maxitive set functions and their possibility distributions

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cramer's theorem.& COPY; 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

Automated summary

This paper introduces a weaker form of maximality and shows that under this assumption, the Shilkret integral is still determined by its possibility distribution for sufficiently regular functions. Inspired by large deviation theory, a Laplace principle for maximative integrals is provided and the possibility distribution is characterized under certain separation and convexity assumptions. Moreover, a maximative integral representation result for weakly maximative non-linear expectations is shown. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities and deriving a monotone analogue of Cramer's theorem.