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 X.
In particular, we show that order closedness, σ(X,X∼
n )-closedness, and
σ(X, L
∞
)-closedness of a law-invariant convex set in X are equivalent,
where X∼
n is the order continuous dual of X. We also provide some application
to proper quasiconvex law-invariant functionals with the Fatou
property.