ORDER CLOSEDNESS OF CONVEX SETS IN BANACH LATTICES

Abstract

Let X be a Banach lattice. An important problem arising from the theory of risk measures asks when order closedness of a convex set in X guarantees closedness with respect to the topology generated by the order continuous dual of X. In this thesis, we present a characterization of when the problem has a positive answer for a wide class of Banach lattices. We also show that under the assumption of law-invariance on the convex set, the problem will always have a positive answer in rearrangement invariant spaces. Motivated by the above problem, similar problems for other types of closedness are also investigated. In the last part of the thesis, we also study several order versions of Banach-Saks properties motivated by the method used by Delbaen and Owari (2019) in solving the above problem for certain Orlicz spaces.