We consider set-valued mappings defined on a topological space with closed
convex images in R-n. The measurability of a multifunction is characterized
by the existence of a Castaing representation for it: a countable set of m
easurable selections that pointwise fills up the graph of the multifunction
. Our aim is to construct a Castaing representation which inherits the regu
larity properties of the multifunction. The construction uses Steiner point
s. A notion of a generalized Steiner point is introduced. A Castaing repres
entation called regular is defined by using generalized Steiner selections.
All selections are measurable, continuous, resp., Holder-continuous, or di
rectionally differentiable, if the multifunction has the corresponding prop
erties. The results are applied to various multifunctions arising in stocha
stic programming. In particular, statements about the asymptotic behavior o
f measurable selections of solution sets via the delta-method are obtained.