The classification problem for nonatomic weak Lp spaces

The aim of the paper is to study the isomorphic structure of the weak Lp space Lp,∞(Ω,Σ,μ) when (Ω,Σ,μ) is a purely nonatomic measure space. Using Maharam’s classification of measure algebras, it is shown that every such Lp,∞(Ω,Σ,μ) is isomorphic to a weak Lp space defined on a weighted direct sum of product measure spaces of the type 2κ . 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 <∞. Applying the methods developed, we obtain an isomorphic classification of the purely nonatomic weak Lp spaces in a special case. © 2009 Elsevier Inc. All rights reserved