  We define two non-linear operations with random (not necessarily closed) sets
in Banach space: the conditional core and the conditional convex hull. While
the first is sublinear, the second one is superlinear (in the reverse set
inclusion ordering). Furthermore, we introduce the generalised conditional
expectation of random closed sets and show that it is sandwiched between the
conditional core and the conditional convex hull. The results rely on
measurability properties of not necessarily closed random sets considered from
the point of view of the families of their selections. Furthermore, we develop
analytical tools suitable to handle random convex (not necessarily compact)
sets in Banach spaces; these tools are based on considering support functions
as functions of random arguments. The paper is motivated by applications to
assessing multivariate risks in mathematical finance.
