On the Riesz dual of ${\bf L^1(\mu)}$

In this article, $(X,\, \mathcal{A},\, \mu)$ is a measure apace. A classical result establishes a Riesz isomorphism between $L^1(\mu)^{\sim}$ and $L^{\infty}(\mu)$ in case the measure $\mu$ is $\sigma$-finite. In general, there still is a natural Riesz homomorphism $\Phi: L^{\infty}(\mu) \to L^1(\mu)^{\sim},$ but it may not be injective or surjective. We prove that always the range of $\Phi$ is an order dense Riesz subspace of $L^1(\mu)^{\sim}$. If $\mu$ is semi-finite, then $L^1(\mu)^{\sim}$ is a Dedekind completion of $L^{\infty}(\mu)$.