Purely non-atomic weak ${\displaystyle L^p}$ spaces

Let (Ω,∑,μ) be a purely non-atomic measure space, and let 1 < p < ∞. If ${\displaystyle L^{p,\infty }(\Omega ,\sum ,\mu )}$ is isomorphic, as a Banach space, to ${\displaystyle L^{p,\infty }(\Omega \prime ,\sum \prime ,\mu \prime )}$ for some purely atomic measure space (Ω',∑',μ'), then there is a measurable partition ${\displaystyle \Omega ={\Omega }_1\cup {\Omega }_2}$ such that ${\displaystyle ({\Omega }_1,\Sigma \cap {\Omega }_1,\mu {\mid }_{\Sigma \cap {\Omega }_1})}$ is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable ${\displaystyle \sigma \subseteq {\Omega }_2}$. In particular, ${\displaystyle L^{p,\infty }(\Omega ,\sum ,\mu )}$ is isomorphic to ${\displaystyle {\ell }^{p,\infty }}$.