On dense univalued representations of multivalued maps (Q1068231)

Let \(\Gamma\) be a measurable multifunction whose domain is a probability space \(\Omega\) and whose values are non-empty closed subsets of a Polish space X. Write \({\mathcal F}\) for the collection of (single-valued) measurable functions from \(\Omega\) to X, and \({\mathcal F}_{\Gamma}\) for the collection of functions in \({\mathcal F}\) which are selectors for \(\Gamma\). The author describes two pseudo-metric topologies on \({\mathcal F}\) with the property that if \(\{f_ n\}\) is dense in \({\mathcal F}_{\Gamma}\), then \(\{f_ n(\omega)\}\) is dense in \(\Gamma\) (\(\omega)\) for almost all \(\omega\in \Omega\).