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 $f:[0,1]\to X$ 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 $\ell_{1}$.