Let $X$ be a Banach space. If $X^{\*\*}$ is linearly isometric to $L\_{1}(\mu)$ for some $\sigma$-finite measue $\mu$, we shall say that $X$ is an $L\_{1}$-pre-bidual.


Question 1. What are the examples of $L\_{1}$-pre-bidual ?


Question 2. Are there any characterizations or even references about $L\_{1}$-pre-biduals ?


Thank you !



Banach spaces whose biduals are $L_{1}$
If $X$ is an infinite dimensional Banach space such that $X^{\*\*}$ is isomorphic to $L^1(\mu)$ for some $\sigma$-finite measure, then $X^{\*\*}$ is non-reflexive, separable and has DPP (Dunford-Pettis property) since reflexivity, separability and DPP are isomorphic properties. This is not possible ([Banach spaces whose second conjugates are separable](https://mathoverflow.net/questions/404226/banach-spaces-whose-second-conjugates-are-separable/404306#404306))


**Edit:** $L^1(\mu)$ is separable when $\mu$ is a $\sigma$-finite measure and $L^1(\mu)$ is a dual Banach space.


a. If $\mu$ is a $\sigma$-finite measure, then $L^1(\mu)$ is isometrically isomorphic to $L^1(\nu)$ for some probability measure, e.g., see *Albiac & Kalton, "Topics in Banach Space Theory", Chapter 5.*


b. If $\mu$ is a probability measure, then $L^1(\mu)$ is a dual space if and only if $\mu$ is purely atomic.


*Proof.* $(\Leftarrow)$ If $\mu$ is purely atomic, then $L^1(\mu)$ is isomorphic to $\ell^1(supp\mu)$, which is a dual space.


$(\Rightarrow)$ $L^1(\mu)$ is weakly compactly generated. Every weakly compactly generated dual Banach space has RNP, and $L^1(\mu)$ has RNP iff $\mu$ is purely atomic; see *Diestel & Uhl, "Vector Measures", Section 7.7.7.*


c. If $\mu$ is a finite purely atomic measure, then $supp\mu$ is countable. Thus, $\ell^1(supp\mu)$ is separable.


