  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 $\ell^{p,\infty}$.
