## Zhou Wei, Zhichun Yang, Jen-Chih Yao, On metrizable vector spaces with the Lebesgue property

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DOI: 10.23952/jnva.6.2022.3.06

Volume 6, Issue 3, 1 June 2022, Pages 239-253

Abstract. In classical analysis, Lebesgue first proved that $\mathbb{R}$ has the Lebesgue property (i.e., each Riemann integrable function from $[a,b]$ into $\mathbb{R}$ is continuous almost everywhere). Though the Lebesgue property may be breakdown in many infinite dimensional spaces including Banach or quasi Banach spaces, to determine spaces with this property is still an interesting issue. This paper is devoted to the study of metrizable vector spaces with the Lebesgue property. As the main results in the paper, we prove that $l^1(\Gamma)$ ($\Gamma$ uncountable) has the Lebesgue property and $\mathbb{R}^{\omega}$, the countable infinite product of $\mathbb{R}$ with itself equipped with the product topology, is a metrizable vector space with the Lebesgue property. In particular, $l^p$, ($1 \textless p\leq+\infty$), as a subspaces of $\mathbb{R}^{\omega}$, is proved to have the Lebesgue property although they are Banach spaces with no such property.