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 κ2. Several isomorphic invariants are then obtained. In particular, it is found that there is a notable difference between the case 1