journal of economic theory 73, 261291 (1997) On Independence for Non-Additive Measures, with a Fubini Theorem Paolo Ghirardato* Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, California 91125 Received May 3, 1995; revised June 21, 1996 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. Journal of Economic Literature Classification Numbers: C44, D81, G10.  1997 Academic Press INTRODUCTION AND MOTIVATION The last four decades have witnessed a steady increase in the usage of non-additive set-functions, rather than probabilities, to represent uncer- tainty. The mathematical theory of non-additive set-functions got its first important contribution with Gustave Choquet's Theory of Capacities [2] in 1953. Choquet's interest was applications to statistical mechanics and potential theory. On the other hand non-additive set-functions started to attract economists' attention after the seminal contribution of Lloyd Shapley [35] (also published in 1953) because of their applications to the study of cooperative games, but the connections with decision theory were not explicitly recognized at that time. It was only with the works of Arthur Dempster (e.g. 13]), later developed by Glenn Shafer [33], that applica- tions to uncertainty and the representation of beliefs were considered. In fact Shafer baptized belief functions the particular set-functions he and article no. ET962241 261 0022-053197 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved. * 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 grate- fully acknowledged. E-mail: paolohss.caltech.edu