Regular Castaing Representations of Multifunctions with Applications to Stochastic Programming

We consider set-valued mappings defined on a topological space with closed convex images in $\mathbb R ^n$. The measurability of a multifunction is characterized by the existence of a Castaing representation for it: a countable set of measurable selections that pointwise fills up the graph of the multifunction. Our aim is to construct a Castaing representation which inherits the regularity properties of the multifunction. The construction uses Steiner points. A notion of a generalized Steiner point is introduced. A Castaing representation called regular is defined by using generalized Steiner selections. All selections are measurable, continuous, resp., Holder-continuous, or directionally differentiable, if the multifunction has the corresponding properties. The results are applied to various multifunctions arising in stochastic programming. In particular, statements about the asymptotic behavior of measurable selections of solution sets via the delta-method are obtained.

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