On closedness of law-invariant convex sets in rearrangement invariant spaces

TL;DR

This paper investigates the closedness properties of law-invariant convex sets in rearrangement invariant spaces, establishing equivalences among various types of closedness and exploring implications for quasiconvex functionals with the Fatou property.

Contribution

It demonstrates the equivalence of different closedness notions for law-invariant convex sets in rearrangement invariant spaces and applies these results to quasiconvex functionals with the Fatou property.

Findings

Order closedness, $\sigma( ext{space}, ext{dual})$-closedness, and $\sigma( ext{space},L^\infty)$-closedness are equivalent for law-invariant convex sets.

Provides conditions under which these closedness properties coincide.

Applications to proper quasiconvex law-invariant functionals with the Fatou property.

Abstract

This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $\sigma(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property.