Distributions of random sets and random selections (Q797202)

A selection of a random set in the complete separable metric space X is defined as a random point belonging to the random set. It is proved that a necessary and sufficient condition for a probability distribution \(\rho\) on X to be a selection of some random set having given probability measure is that, for all closed \(C\subset X\), \(\rho\) (C) is bounded above by the probability of the random set having non-empty intersection with C. The inequalities are analogous to those required for the existence of matchings in the so-called marriage problem. The family of distributions which may arise as selections of a random set measure is convex and compact according to the Prohorov metric, and depends continuously upon the random set measure.