The largest class of closed convex valued multifunctions for which Effros measurability and scalar measurability coincide
The paper examines measurability notions for set-valued maps which take values in the space of all closed convex subsets of a normed linear space. Of interest here, in particular, are the Effros measurability, namely measurability with respect to the Effros \(\sigma\)-field of sets, and scalar measurability, namely measurability of all support functionals applied to the set-valued map. An example demonstrates that the latter notion is weaker than the Effros measurability. Then the paper introduces the notion of a countably supported family of sets; this, roughly, means that each set is determined by a prescribed denumerable collection of support functionals. If the set-valued map takes values in a countable supported subspace, and if, in addition, every Borel set is analytic, then, so the paper verifies, the scalar measurability and the Effros measurability coincide. The paper offers enlightening examples and comments on the extent of the new notion, and the applicability of the main result.
scientific article; zbMATH DE number 1268113