In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L\_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L\_{1}$ such that $X$ has the Schur property, but $X$ is not isomorphic to a subspace of $l\_{1}$. Hence, I have the first question as follows:


Question 1. Is every Banach space with the Schur property isomorphic to a subspace of $L\_{1}(\mu)$ for some measure $\mu$?


Moreover, W. B. Johnson and E. Odell gave natural non-trivial conditions that a subspace of $L\_{p}$ embeds into $l\_{p}$ for $1<p<\infty, p\neq 2$. But 


Question 2. Are there conditions that a subspace of $L\_{1}$ embeds into $l\_{1}$ ?


Thank you!



Can every Banach space with the Schur property embed into $L_{1}(\mu)$ for some $\mu$?
Absolutely not. Take the the $\ell\_1$-sum of $\ell\_\infty^n$ ($n\in \mathbb N$). If that embedded into $L\_1(\mu)$, then you would have found $c\_0$ in some ultrapower of $L\_1(\mu)$, which is impossible.


