Purely non-atomic weak L^p spaces

Let $\msp$ be a purely non-atomic measure space, and let $1 < p < \infty$. If $\weakLp\msp$ is isomorphic, as a Banach space, to $\weakLp\mspp$ for some purely atomic measure space $\mspp$, then there is a measurable partition $Ω= Ω_1\cupΩ_2$ such that $(Ω_1,Σ\capΩ_1,μ_{|Σ\capΩ_1})$ is countably generated and $σ$-finite, and that $μ(σ) = 0$ or $\infty$ for every measurable $σ\subseteq Ω_2$. In particular, $\weakLp\msp$ is isomorphic to $\ell^{p,\infty}$.

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