On the Lebesgue Property of Monotone Convex Functions

TL;DR

This paper characterizes the Lebesgue property of monotone convex functions on solid vector spaces using duality and weak inf-compactness, unifying recent results in convex risk measures.

Contribution

It provides a unified characterization of order-continuity for monotone convex functions via dual space properties, extending previous work in convex risk measures.

Findings

Characterization of Lebesgue property via weak inf-compactness

Dual attainment of supremum by order-continuous linear functionals

Unification of recent results in convex risk measure theory

Abstract

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous linear functionals. This generalizes and unifies several recent results obtained in the context of convex risk measures.