The classification problem for nonatomic weak [formula omitted] spaces

Abstract

The aim of the paper is to study the isomorphic structure of the weak L p space L p , ∞ ( 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 L p , ∞ ( Ω , Σ , μ ) is isomorphic to a weak L p space defined on a weighted direct sum of product measure spaces of the type 2 κ . 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 L p spaces in a special case.