Second dual projection characterizations of three classes of L0-closed, convex, bounded sets in L1

Let τλ be the topology of convergence locally in measure on L1=L1(λ) and P be the Yosida–Hewitt projection from onto L1. We characterize convex, τλ-compact subsets C of L1 as precisely those for which P is a compactness preserving map from with the weak∗-topology to C with the τλ-topology. We further show that a convex, τλ-closed, L1-norm bounded subset C of L1 is a Schur set if and only if is sequentially continuous. Finally, we discover which τλ-closed, bounded, convex subsets C of L1 are such that is continuous. We call such sets C good. They turn out to be precisely the pluriweak-to-measure-continuity sets, in the sense defined below.