The classification problem for nonatomic weak Lp spaces

The aim of the paper is to study the isomorphic structure of the weak L-p space L-p,L-infinity(Omega, Sigma, mu) when (Omega, Sigma, mu) is a purely nonatomic measure space. Using Maharani's classification of measure algebras, it is shown that every such L-p,L-infinity(Omega, Sigma, mu) is isomorphic to a weak L-p space defined on a weighted direct sum of product measure spaces of the type 2(kappa). Several isomorphic invariants are then obtained. In particular, it is found that there is a notable difference between the case 1 < p < 2 and the case where 2 <= p < infinity. Applying the methods developed, we obtain an isomorphic classification of the purely nonatomic weak LP spaces in a special case. (C) 2009 Elsevier Inc. All rights reserved.