p非原子弱L^p空间

摘要：

设$\msp$是一个纯非原子测度空间，令$1 < p < \infty$。如果作为Banach空间，$\weakLp\msp$同构于对于某个纯原子测度空间$\mspp$的$\weakLp\mspp$，则存在可测划分$Ω= Ω_1\cupΩ_2$，使得$(Ω_1,Σ\capΩ_1,μ_{|Σ\capΩ_1})$是可数生成的，并且$σ$是有限的，并且对于每个可测$σ\subseteq Ω_2$，$μ(σ) = 0$或$\infty$。特别地，$\weakLp\msp$与$\ell^{p,\infty}$同构。

英文摘要：

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 $Ω= Ω_1\cupΩ_2$ such that $(Ω_1,Σ\capΩ_1,μ_{|Σ\capΩ_1})$ is countably generated and $σ$-finite, and that $μ(σ) = 0$ or $\infty$ for every measurable $σ\subseteq Ω_2$. In particular, $\weakLp\msp$ is isomorphic to $\ell^{p,\infty}$.

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