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 $\Omega = \Omega_1\cup\Omega_2$ such that $(\Omega_1,\Sigma\cap\Omega_1,\mu_{|\Sigma\cap\Omega_1})$ is countably generated and $\sigma$-finite, and that $\mu(\sigma) = 0$ or $\infty$ for every measurable $\sigma \subseteq \Omega_2$. In particular, \weakLp\msp is isomorphic to $\ell^{p,\infty}$