Products

Abstract

In probability theory, the generalized product of measures is a constructive way for introducing conditional expectation. The abstract definition of conditional expectation essentially uses the Radon-Nikodym Theorem, so it works only for the σ-additive case. Hence, for non additive set functions one might be interested to know what can be achieved through the constructive way. But even here a part of Fubini’s Theorem, namely that the integral with respect to the product of the set functions equals the repeated integral, remains valid only if the set function for the second integration is additive. More general, for submodular set functions one only gets an inequality: The repeated integral does not exceed the integral. Thus the problem of generalizing conditional expectation beyond the a-addtive case remains open to a large extent. For an alternative but likewise incomplete approach see Denneberg 1994. At least we prove in this chapter the full Theorem of Fubini for measures and also for (finitely) additive set functions.