On the parametrized integral of a multifunction: the unbounded case
Consider a measure space \((T,\tau,\lambda)\) with \(\lambda\) positive \(\sigma\)-finite, a measurable space \((\Omega,{\mathfrak A})\) such that \(\tau\) and \({\mathfrak A}\) are countably generated, a transition probability \(\beta: T\times{\mathfrak A}\to [0,a]\), a separable Banach space \(Y\), a \(\tau\otimes{\mathfrak A}\)-measurable closed valued multifunction \(F: T\times\Omega\to 2^Y\) such that \(F(t,\cdot)\) is \(\beta(t,\cdot)\)-integrable for \(\lambda\)-almost all \(t\in T\). Assume further that there exist a \(\lambda\)-null set \(N\) and a probability measure \(Q\) on \((\Omega,{\mathfrak A})\) such that \(Q\) dominates \(\beta(t,\cdot)\) for all \(t\in T\setminus N\). The author states the following selection theorem. Theorem A: (a) The multifunction \(G:t \mapsto\int_\Omega F(t,\omega)\beta_t(d\omega)\) is \(\tau\)-mesurable. (b) If \(g: T\to Y\) is \(\tau\)-measurable such that \(g(t)\in G(t)\) for \(\lambda\)-almost \(t\in T\), then there exists a \(\tau\otimes{\mathfrak A}\)-measurable selection \(f: T\times\Omega\to Y\) of \(F\) such that for \(\lambda\)-almost all \(t\in T\), \(f(t,\cdot)\in L^1_Y(\Omega,{\mathfrak A},\beta_t)\) and satisfies \(g(t)= \int_\Omega f(t,\omega)\beta_t(d\omega)\) for almost all \(t\in T\). A variant of the preceding result concerns the approximate representation of a measurable selection of \(\text{cl}(G(t))\) with the assumption: \(F\) is not necessarily closed valued and is graph measurable, that is, \(Gr(F)\in \tau\times{\mathfrak A}\otimes\beta(Y)\), \(\beta(Y)\) being the Borel tribe of \(Y\). Theorem B: (a) \(G\) is \(\tau_X\)-measurable; (b) if \(h\) is a \(\tau_X\)-measurable selection of \(\text{cl}(G)\), then there exists a sequence \((f_n)_{n> 1}\) of \(\tau\otimes{\mathfrak A}\)-measurable selections of \(F\) such that for all \(N\geq 1\) and for \(\lambda\)-almost all \(t\in T\), \(f_n(t,\cdot)\in L^1_Y(\Omega,{\mathfrak A}\beta_t)\) and \(h(t)= \lim_{n\to\infty}\, \int_\Omega f_n(t,\omega) \beta_t(d\omega)\) for \(\lambda\)-almost all \(t\in T\), when the limit is taken in the norm topology of \(Y\). An inspection of some proofs given here should lead to other variants. See for instance the proof of Proposition 2.7.
scientific article; zbMATH DE number 5146251
scientific article