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arXiv · 2303.01434

Banach Spaces with the Lebesgue Property of Riemann Integrability

Harrison Gaebler·Bunyamin Sari
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Abstract

A Banach space is said to have the Lebesgue property if every Riemann-integrable function $f:[0,1]\to X$ is Lebesgue almost everywhere continuous. We give a characterization of the Lebesgue property in terms of a new sequential asymptotic structure that is strictly between the notions of spreading and asymptotic models. We also reproduce an apparently lost theorem of Pelczynski and da Rocha Filho that a subspace $X\subset L_{1}[0,1]$ has the Lebesgue property if every spreading model of $X$ is equivalent to the unit vector basis of $\ell_{1}$.

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Harrison Gaebler, Bunyamin Sari. 2023-03-02. Banach Spaces with the Lebesgue Property of Riemann Integrability. https://arxiv.org/abs/2303.01434

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