论文标题
在$ {\ bf l^1（μ）} $的riesz dual上
On the Riesz dual of ${\bf L^1(μ)}$
论文作者
论文摘要
在本文中，$（x，\，\ Mathcal {a}，\，μ）$是量度的速度。一个经典的结果建立了$ l^1（μ）^{\ sim} $和$ l^{\ infty}（μ）$之间的riesz同构。通常，仍然有天然的riesz同构$φ：l^{\ infty}（μ）\至l^1（μ）^{\ sim}，但可能不是注射式或过度的。我们证明，$φ$的范围始终是$ l^1（μ）^{\ sim} $的订单密集riesz子空间。如果$μ$是半金融的，则$ l^1（μ）^{\ sim} $是$ l^{\ infty}（μ）$的Dedekind完成。
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 $L^{\infty}(μ)$.
