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