On Riemann Integration in Metrizable Vector Spaces

Abstract

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 $\mathbb{R}^{\omega}$, 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 $\mathbb{R}^{\omega}$, possess the Lebesgue property although they are Banach spaces having no such property.