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

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 $\sigma (\mathcal{X},L^{\mathrm{\infty }})$-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.