Towards a characterization of the property of Lebesgue (Q2054574)

A real Banach space \(X\) is said to have the property of Lebesgue (abbr. PL), or to be a PL-space, if every Riemann integrable function \(f\colon[0,1]\to X\) is necessarily almost everywhere continuous (like in the case \(X=\mathbb R\)). All finite-dimensional spaces are PL-spaces, moreover so is \(\ell_1\). On the other hand, no \(\ell_p\) space has PL for \(p\in(1,\infty]\). One of the most discussed problems in Banach space geometry is that of characterising PL-spaces. The article under review contributes to this discussion with certain sufficient/necessary conditions related to the asymptotic structure of \(X\). The main results are the content of Section 3. First, in Theorem 1.0.2, the author generalises \textit{K. M. Naralenkov}'s result [Real Anal. Exch. 33, No. 1, 111--124 (2008; Zbl 1151.26008)], which says that all asymptotic-\(\ell_1\) spaces have PL, to the coordinate free case. In Theorem 1.0.4 certain uniqueness property of asymptotic models in PL-spaces is revealed. The third main result, Theorem 1.0.5, demonstrates a setting in which an asymptotic-\(\ell_1\)-like condition on \(X\) becomes necessary for PL. This all is preceded, in Section 1 and 2, by an introduction to PL-spaces, where, in particular, the property of Lebesgue is described in terms of the so-called Darboux integral.