Let $(X,\Sigma,\mu)$ be a measure space. Then $L^1(X,\Sigma,\mu)$ is a Banach space. I am wondering when $L^1(X,\Sigma,\mu)$ is isomorphic to the (continuous) dual space $Y^\ast$ of some Banach space $Y$, where isomorphic means there exists a continuous invertible linear map (not necessarily isometric) between the 2 spaces. Here are some special cases I am aware of:
If $X$ is finite then $L^1(X,\Sigma,\mu)$ is a finite dimensional vector space, hence reflexive.
If $X = \mathbb{N}$ and $\mu$ is counting measure, then $L^1(X,\Sigma,\mu) = l^1(\mathbb{N}) \cong c_0^\ast$.
Beyond that I don't really have any other examples. I don't even know of an example of a measure space such that $L^1(X,\Sigma,\mu)$ is not isomorphic to a dual space. The closest thing I can think of is $X = \mathbb{R}$ and $\mu =$ Lebesgue measure, and then $L^1(X,\Sigma,\mu)$ is not reflexive. Any input is appreciated.
