Continuous selections of Aumann integrals (Q916812)

Let K be a lower semicontinuous multifunction defined on a locally compact Polish space, whose values are closed decomposable subsets of the Lebesgue space \(L^ 1.\) It is proved, that for every continuous selection of the Aumann integral of K and for every \(\epsilon >0\) there exists a continuous selection k of K such that the uniform distance between the integral of k and the given selection of the Aumann integral of K is smaller than \(\epsilon\). Some interpretations of the above result for a family of vector measures and a theorem concerning Carathéodory's selections are also included.