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.
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