Max-min (σ-)additiye representation of monotone measures

In non-additive measure and integration (or fuzzy measure and integral) one tries to generalise the issues of product measure and conditional expectation from the additive theory. In the discrete case successful attempts have been made via the max-min additive representation of the monotone measure and the corresponding integrals. The present paper intends to find, for arbitrary monotone measures, a max-min additive representation and, under certain topological assumptions, a representation with σ-additive measures, thus providing a powerful tool for the theory of non-additive measure and integration. Received: December 2000; revised version: September 2001