The classification problem for nonatomic weak Lp spaces

Abstract

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.