可度量化向量空间中的Riemann积分
On Riemann Integration in Metrizable Vector Spaces
摘要：
在经典分析中，Lebesgue首先证明了$\mathbb{R}$具有从$[a,b]$到$\mathbb{R}$的每个Riemann可积函数几乎处处连续的性质。这个性质被称为勒贝格性质。虽然Lebesgue性质在许多无限维空间包括Banach空间和拟巴拿赫空间中都可能被破坏，但如何确定具有这一性质的空间仍然是一个有趣的问题。本文研究了可度量化向量空间中向量值函数的Riemann积分，证明了连续函数的微积分和基元基本定理。进一步证明了具有积拓扑的$\mathbb{R}$的可数无穷乘积$\mathbb{R}^ω$是具有Lebesgue性质的可度量化向量空间，并证明了作为$\mathbb{R}^ω$的子空间的$l^p(1<p\leq+\infty)$具有Lebesgue性质，尽管它们是不具有Lebesgue性质的巴拿赫空间.
英文摘要：
In classical analysis, Lebesgue first proved that $\mathbb{R}$ has the property that each Riemann integrable function from $[a,b]$ into $\mathbb{R}$ is continuous almost everywhere. This property is named as the Lebesgue property. Though the Lebesgue property may be breakdown in many infinite dimensional spaces including Banach or quasi Banach spaces, to determine spaces having this property is still an interesting problem. In this paper, we study Riemann integration for vector-value functions in metrizable vector spaces and prove the fundamental theorems of calculus and primitives for continuous functions. Further we discovery that $\mathbb{R}^ω$, the countable infinite product of $\mathbb{R}$ with itself equipped with the product topology, is a metrizable vector space having the Lebesgue property and prove that $l^p(1<p\leq+\infty)$, as subspaces of $\mathbb{R}^ω$, possess the Lebesgue property although they are Banach spaces having no such property.
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