On Independence For Non-Additive Measures, With a Fubini Theorem
Recent models of decision making represent agents' beliefs by non-additive set-functions. An important technical question which arises in applications to diverse areas of economics is how to define independence of such set-functions. After arguing that the straightforward generalization of independence does not in general yield a unique product, in this work I show that, while Fubini's theorem is in general false if additivity is not granted, it is true when a certain type of function is being integrated. For these functions the iterated integrals coincide with the integral with respect to products which satisfy a certain property, strictly stronger than independence. I show that most of the assumptions made in these results are very close to being necessary. In general the mentioned property is still not strong enough to uniquely define a product. On the other hand I discuss some proposals which have been made in the literature, and I show that unicity can however be obtained when the product is assumed to be a belief function. Moreover I show that the unique product thus obtained has an intuitive justification when the marginals are distributions induced by random correspondences. Finally I use the results in the paper to discuss the question of randomization in decision models with non-additive beliefs.