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Banach Spaces with the Lebesgue Property of Riemann Integrability

Harrison Gaebler, Bunyamin Sari

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 $\ell_{1}$.

Banach Spaces with the Lebesgue Property of Riemann Integrability

Abstract

A Banach space is said to have the Lebesgue property if every Riemann-integrable function 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 has the Lebesgue property if every spreading model of is equivalent to the unit vector basis of .
Paper Structure (8 sections, 9 theorems, 60 equations)

This paper contains 8 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Background Material
  3. Riemann Integration in Banach Spaces
  4. Haar systems of partitions of $\mathbb{N}$
  5. A Characterization of Banach spaces with the Lebesgue property
  6. Asymptotic structures and Haar-$\ell^+ _1$ sequences
  7. Argyros-Motakis space $X_{iw}$
  8. The $L_{1}[0,1]$ Theorem of Pelczynski and da Rocha Filho

Key Result

Proposition 2.2

Let $f:[0,1]\to X$ be given. Then, $f\in\mathcal{R}([0,1],X)$ if and only if there exists a vector $x_{f}\in X$ such that, for every $\varepsilon>0$, there is a positive integer $n=n(\varepsilon)\in\mathbb{N}$ so that $\|x_{f}-\frac{1}{2^{m}}\sum_{i=1}^{2^{m}}f(t_{i})\|<\varepsilon$ for every $m\geq

Theorems & Definitions (23)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • proof : Proof of Proposition \ref{['Haydon-Odell']}
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • ...and 13 more