Parametrized integral of multifunctions in Banach spaces
Let \(T\), \(S\) be complete separable metric spaces, \(E\) a separable Banach space, \(\lambda\) a finite measure on \(T\), \((\beta_t)_{t\in T}\) a family of probability measures on \(S\), and let \(F\) be a measurable multifunction from \(T\times S\) to the space of all nonempty compact subsets of \(E\). Suppose further that for all \(t\), the multifunction \(s\to F(t,s)\) is \(\beta_t\)-integrably bounded and for any \(D\in \mathbb{B}(s)\), \(t\to \beta_t(D)\) is Borel-measurable. It is asserted then that the multifunction \[ t\to \Gamma(t)= \text{cl}\Biggl(\int_s F(t,s) \beta_t(ds)\Biggr) \] is \(\mathbb{B}_\lambda(T)\)-measurable. Moreover, the structure for measurable selections for \(\Gamma\) in terms of integrals of a sequence of measurable selections of \(F\) is obtained. It is asserted that when \(F\) is convex-valued one can find a function \(g\) such that, if \(\gamma\) is a selection of \(\Gamma\), \[ \gamma(t)= \int_s g(t,s)\beta_t(ds). \]
scientific article; zbMATH DE number 1389235