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