Purely non-atomic weak L^p spaces

Abstract

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