Weak convergence of probability measures on hyperspaces with the upper Fell-topology (Q6645213)

Let \(\mathcal{F}\) and \(\mathcal{K}\) the families of closed and compact sets, respectively, in the topological space \((E,\mathcal{G})\) and, for an arbitrary subset \(A\) of \(E\), define\N\[\N\mathcal{M}(A):=\{F\in\mathcal{F}:F\cap A=\emptyset\}\quad \mathcal{H}(A):=\{F\in\mathcal{F}:F\cap A\ne\emptyset\}\,,\N\]\Nand consider the topologies \(\tau_F\) and \(\tau_{uF}\) on \(\mathcal{F}\) generated by\N\[\N\mathcal{S}_F=\{\mathcal{M}(K):k\in\mathcal{K}\}\cup\{\mathcal{H}(G):G\in\mathcal{G}\}\N\]\Nand by\N\[\N\mathcal{S}_{uF}=\{\mathcal{M}(K):k\in\mathcal{K}\}\N\]\Nrespectively; they are the \textit{Fell-topology} and \textit{the upper Fell-topology}, see [\textit{J. M. G. Fell}, Proc. Am. Math. Soc. 13, 472--476 (1962; Zbl 0106.15801)]. The main object of the paper is about characterisations of \emph{weak convergence} in the sense of \textit{F. Topsoe} [Topology and measure. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0197.33301)] on \((\mathcal{F},\tau_F)\). Since \(\tau_F\) is stronger than \(\tau_{uF}\) weak convergence in \((\mathcal{F},\tau_F)\) implies that on \((\mathcal{F},\tau_{uF})\); the converse holds under additional conditions, see Theorem 2.9. A connexion is also found with the \textit{upper Vietoris topology}, in which \(\mathcal{F}\) replaces \(\mathcal{K}\) in \(\tau_{uF}\). A surprising result is that every net of probability measures converges weakly in \((\mathcal{F},\tau_{uF})\). The results are extended to random closed sets in \(E\) and measurable selections.