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Abstract

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If $L^{p,∞}(Ω,∑,μ)$ is isomorphic, as a Banach space, to $L^{p,∞}(Ω',∑',μ')$ for some purely atomic measure space (Ω',∑',μ'), then there is a measurable partition $Ω = Ω_{1} ∪ Ω_{2}$ such that $(Ω_{1}, Σ ∩ Ω_{1},μ|_{Σ ∩ Ω_{1}})$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable $σ ⊆ Ω_{2}$. In particular, $L^{p,∞}(Ω,∑,μ)$ is isomorphic to $ℓ^{p,∞}$.