Extremal points without compactness in L^1(\mu)

We investigate the existence of extremal points and the Krein-MIlman representation A = co ExtA of bounded convex subsets of L^1(μ) which are only closed with respect to the topology of μ-a.e. convergence. In particular we prove existence for non-empty, bounded and convex subsets of L^1(μ) that are μ-closed and order-bounded from below. As for the representation of a set as A = co ExtA it is shown that it holds if the DA faces of A are μ-closed. Finally we consider Kuratowski sum of sets and deduce some further properties of extremal structure for sets having μ-closed extremal faces.