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- Title
ON METRIZABLE VECTOR SPACES WITH THE LEBESGUE PROPERTY.
- Authors
ZHOU WEI; ZHICHUN YANG; JEN-CHIH YAO
- Abstract
In classical analysis, Lebesgue first proved that R has the Lebesgue property (i.e., each Riemann integrable function from [a,b] into 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 l¹(ܓ) (ܓ uncountable) has the Lebesgue property and Rὠ, the countable infinite product of R with itself equipped with the product topology, is a metrizable vector space with the Lebesgue property. In particular, lp, (1 < p<+∞), as a subspaces of Rɷ, is proved to have the Lebesgue property although they are Banach spaces with no such property.
- Subjects
VECTOR spaces; LEBESGUE integral; MATHEMATICS theorems; BANACH spaces; RIEMANN integral
- Publication
Journal of Nonlinear & Variational Analysis, 2022, Vol 6, Issue 3, p239
- ISSN
2560-6921
- Publication type
Article
- DOI
10.23952/jnva.6.2022.3.06