**Sankhya:
The Indian Journal of Statistics**

2004, Volume 66, Pt. 3, 409--427

**Random Correspondences as Bundles
of Random Variables**

By

Adriana Castaldo, University of Sussex,
Brighton, U.K.

Fabio Maccheroni, Universit{\'a} Bocconi, Milano, Italy

Massimo Marinacci, Universit{\'a} di Torino, Torino, Italy

SUMMARY. We relate the distributions induced by random sets with the distributions induced by their measurable selections, thus providing a probabilistic foundation for viewing random sets as ``sets'' of random variables. In so doing we revisit and extend some results of Artstein, Hart, Hess, and Kohlberg, and we obtain a ``change of variable formula'' for the Aumann integral of a random set.

*AMS (1991) subject classification}. ***28B20, 28A12, 54C60, 54C65,
60D05**.

*Key words and phrases. *Distributions of random sets, measurable
selections, Choquet integral, Aumann integral.