On Independence for Non-Additive Measures, with a Fubini Theorem
An important technical question arising in economic and financial applications of decision models with non-additive beliefs is how to define stochastic independence. In fact the straightforward generalization of independence does not in general yield a unique product. I discuss the problem of independence, with specific focus on the validity of the Fubini theorem. The latter holds in general only for a special class of functions. It also requires a stronger notion of independent product. This is unique when the product must be a belief function. Finally I discuss an application to the issue of randomization in decision making.
© 1997 Academic Press. Received 3 May 1995, Revised 21 June 1996. This paper is a modified version of Chapter 4 of my doctoral dissertation at UC Berkeley. I thank my adviser Bob Anderson, Massimo Marinacci, Klaus Nehring, Chris Shannon, and especially Marco Scarsini for helpful comments and discussion. Detailed comments from a referee and an associate editor greatly helped making this paper leaner. The usual disclaimer applies. Financial support from an Alfred P. Sloan Doctoral Dissertation Fellowship is gratefully acknowledged.