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]→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⊂L1[0,1] has the Lebesgue property if every spreading model of X is equivalent to the unit vector basis of ℓ1.