Just Previsions

2026-05-11

AI Summary

This study addresses the structural representation of general previsions, moving beyond traditional frameworks confined to sublinear or superlinear previsions. By leveraging tools from functional analysis, topology, and order-theoretic structures, the work establishes—for the first time—a homeomorphic relationship between the space of general previsions and the spaces of sublinear (or superlinear) previsions. This construction is characterized via orthogonality relations, yielding a novel dual power-space structure. The main contribution lies in proving that, under mild conditions, any prevision can be expressed as the infimum of a family of sublinear previsions or the supremum of a family of superlinear previsions, thereby achieving an extremal decomposition and a structural characterization of general previsions.

Abstract

Previsions are positively homogeneous functionals, and are generalized forms of integration functionals. We investigate previsions -- just previsions, not sublinear or superlinear previsions as in previous work. We show that every prevision can be expressed as an infimum of sublinear previsions, and as a supremum of superlinear previsions under mild conditions. This extends to homeomorphisms between spaces of previsions and certain hyperspaces over spaces of sublinear or superlinear previsions, which can also be characterized in terms of orthogonality relations, making the construction a variant of a double powerspace construction.

Problem

• previsions
• sublinear previsions
• superlinear previsions
• hyperspaces
• orthogonality

Innovation

• previsions
• sublinear previsions
• superlinear previsions
• hyperspaces
• double powerspace