Geometric properties of universally measurable mappings (Q2773591)

The author studies relations between the so-called universally measurable mappings and geometrically-topological properties of those mappings. In particular, mappings are considered of the \({\mathcal D}\)-class introduced by the author in [Math. Notes 26, No. 6, 979-982 (1980); translation from Mat. Zametki 26, 949-955 (1979; Zbl 0441.28004)]. Emphasis is given to the following problem: Find a family \({\mathcal A}\) of mappings from a separable topological space \(S\) to a metric space \(R\) such that the following statement holds true: \(f\in {\mathcal A}\) is a universally measurable mapping if and only if \(f\in {\mathcal D}\).NEWLINENEWLINENEWLINEA particular case of this problem was studied in the above-mentioned article, where the author proved that if \(T\) is a linear continuous operator acting in Banach spaces \(X\) and \(Y\) then the mapping \(T^*\) is universally measurable from \((Y^*,w^*)\) to \((X^*,\|{\cdot}\|)\), with \(w^* = \sigma(Y^*,Y)\), if and only if \(T^* \in {\mathcal D}((Y,w^*),X^*)\).NEWLINENEWLINENEWLINEIn the article under review, the author generalizes the above result to the case in which \(S\) and \(R\) are \(^*\)-weakly compact subsets of \(Y^*\) and \(X^*\) and proves that, for every \(^*\)-weakly continuous mapping from \(S\) to \(R\), universal measurability of a mapping \(f\) from \((S,w^*)\) to \((R,\|{\cdot}\|)\) means that \(f \in {\mathcal D}((S,w^*),(R,\|{\cdot}\|))\). This result is based on the author's theorem presenting a new characterization of universally measurable mappings. Using the obtained results, the author exposes a new series of properties for universally measurable mappings.