Maximum Lebesgue Extension of Monotone Convex Functions

TL;DR

This paper constructs the largest possible extension of a monotone convex function with the Lebesgue property from bounded to more general random variables, using explicit methods and Orlicz-type spaces.

Contribution

It introduces a maximum extension of monotone convex functions with the Lebesgue property, characterized by a specific uniform integrability condition and explicit construction.

Findings

Existence of a maximum Lebesgue extension with explicit construction

Characterization of Lebesgue property via uniform integrability

Dual representation of the extended function

Abstract

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function.