If a closed valued (resp. compact valued) lsc (resp. usc) multifunction acts from a subset \(A\) of a metrizable space to subsets of a Polish space, then it can be extended to a \(G_\delta\)-set \(B\) containing \(A\), preserving assumptions concerning the type of values and semicontinuity.
Random versions of theorems of this kind, concerning lower and upper Carathéodory multifunctions of two variables, are proved. The case of Banach space-valued multifunctions, Borel measurability and existence of Carathéodory’s multiselections are also treated.