To answer the question in the title, there is the following general theorem:

The natural map $I \colon L^\infty(X) \to (L^1(X))^\ast$ is

- an isometric injection if and only if $(X,\Sigma,\mu)$ is semifinite.
- an isometric isomorphism if and only if $(X,\Sigma,\mu)$ is localizable.

You already covered point 1 in your question (the isometry property being straightforward). The second point is more delicate.

A measure space $(X,\Sigma,\mu)$ is called *localizable* if it is semifinite and in addition the following condition holds:

For every family $\mathcal{F} \subseteq \Sigma$ there is $H \in \Sigma$ such that

- $F \setminus H$ is a null set for all $F \in \mathcal{F}$.
- If $G \in \Sigma$ is such that $F \setminus G$ is a null set for all $F \in \mathcal{F}$ then $H \setminus G$ is a null set.

Loosely speaking, this property asserts that every family $\mathcal{F}$ of measurable sets has a smallest measurable envelope $H$ (up to null sets).

The definition of localizability implies via a slightly technical argument that one can glue together measurable functions. More precisely:

Let $\mathscr{F}$ be a family of functions such that each $f \in \mathscr{F}$ is defined and measurable on a measurable subset $D_f$ of the localizable measure space $(X,\Sigma,\mu)$. Suppose in addition that $f_1 = f_2$ a.e. on $D_{f_1} \cap D_{f_2}$ whenever $f_1,f_2 \in \mathscr{F}$. Then there is a measurable function $g \colon X \to \mathbb{R}$ whose restriction to $D_f$ satisfies $g|_{D_f} = f$ a.e. for all $f \in \mathscr{F}$.

Using this result one can patch together the Radon-Nikodym derivatives one obtains from restricting a continuous linear functional $\varphi \colon L^1(X) \to \mathbb{R}$ to the subspaces $L^1(F) \subseteq L^1(X)$ where $F$ runs through the subsets of finite measure of $X$.

The proof of the converse direction proceeds by a direct verification of the localizability property of $(X,\Sigma,\mu)$ the fact that $I$ is isometric implies semi-finiteness by point 1. of the theorem and the "envelope condition" uses surjectivity of $I$.

Details can be found in 243G on page 153 of this PDF. The gluing property is proved in 213N on page 28.