论文标题
可测量函数的向量晶格的表征
A Characterization of the Vector Lattice of Measurable Functions
论文作者
论文摘要
考虑到概率度量空间$（x，σ，μ）$，众所周知，Riesz Space $ l^0（μ）$的等价类别的可测量函数$ f：x \ to \ mathbf {r} $是普遍完整的，恒定功能$ \ sathbf {1} $是一个弱订单单位。此外，由$ f \ mapsto \ int f \，\ mathrm {d}μ$定义的线性函数$ l^\ infty（μ）\ to \ mathbf {r} $严格呈正，并且连续订购。在这里，我们特别表明，匡威（Converse）是正确的，即，与较弱的订单单位$ e> 0 $相关的任何普遍完成的Riesz Space $ e $，该$ e> 0 $承认，该订单严格正订单连续线性在$ e $中产生的主要理想是lattice Is lattice Isomorphic iS iSomorphic insomorphic in $ l^0（μ）$，以某种概率的量度测量级$（x，x，x，x ,, x，n）。
Given a probability measure space $(X,Σ,μ)$, it is well known that the Riesz space $L^0(μ)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a weak order unit. Moreover, the linear functional $L^\infty(μ)\to \mathbf{R}$ defined by $f \mapsto \int f\,\mathrm{d}μ$ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space $E$ with a weak order unit $e>0$ which admits a strictly positive order continuous linear functional on the principal ideal generated by $e$ is lattice isomorphic onto $L^0(μ)$, for some probability measure space $(X,Σ,μ)$.
