Purely non-atomic weak Lp spaces

Abstract

Let (Ω, Σ, μ) be a purely non-atomic measure space, and let 1 < p < ∞. If Lp,∞(Ω, Σ, μ) is isomorphic, as a Banach space, to Lp,∞(Ω′, Σ′, μ′) for some purely atomic measure space (Ω′, Σ′, μ′), then there is a measurable partition Ω = Ω1 ∪ Ω2 such that (Ω1, Σ ∩ Ω1, μ|Σ∩Ω1) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω2. In particular, Lp,∞(Ω, Σ, μ) is isomorphic to ℓp,∞.