On the integrable selections of certain multifunctions (Q1917683)

Let \(F:T\to E\) be a closed valued measurable multifunction from a nonatomic \(\sigma\)-finite measure space \(T\) into a separable Banach space \(E\). Conditions are formulated under which the set of all integrable selections of \(F\) intersects each member of a specially defined family \(\mathfrak V\) of subsets of \(L^1(T,E)\). Among examples of \(\mathfrak V\) there is a family of all closed hyperplanes of \(L^1\). There are indicated many consequences, e.g., concerning relations between convergence of sequences of multifunctions and its selectors, further, properties of Aumann integral, Gâteaux differentiability of the Nemytski operator, etc.