Just Previsions

TL;DR

This paper explores the structure of previsions, a class of positively homogeneous functionals, showing their representation as infima and suprema of sublinear and superlinear previsions, respectively, and establishing related homeomorphisms.

Contribution

It provides a novel characterization of previsions as infima and suprema of sublinear and superlinear previsions, extending the understanding of their structure and relationships.

Findings

Every prevision can be expressed as an infimum of sublinear previsions.

Every prevision can be expressed as a supremum of superlinear previsions.

Homeomorphisms between spaces of previsions and hyperspaces are characterized by orthogonality relations.

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.