Asymptotic properties and Riemann integration (Q2371915)

The author describes the class of Banach spaces in which the Lebesgue criterion of Riemann integrability is valid. More precisely, a real Banach space \(X\) is said to have the Lebesgue property, if any Riemann integrable function from the closed unit interval \([0,1]\) into \(X\) is continuous almost everywhere on \([0,1]\). In 1991, Gordon published the first truly non-classical example of a Lebesgue space: the Tsirelson space \(T\). \(T\), being close to \(l^1\) in an asymptotic sense, is reflexive and does not contain an isomorphic copy of either \(l^p\) or \(c_0\). This paper extends Gordon's result to prove that an asymptotic \(l^1\) space has the Lebesgue property.