The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem and the Krein-Šmulian theorem are shown.

This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of L 0 -modules. In the first part of the paper we develop the general theory and provide a robust representation of conditional assessment indices, and in the second part we apply this theory to dynamic acceptability indices acting on stochastic processes.

The classical Brouwer fixed point theorem states that in R d every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L 0 = L 0 ( , A, P) be the set of random variables. We consider (L 0 ) d as an L 0 -module and show that local, sequentially continuous functions on L 0 -convex, closed and bounded subsets have a fixed point which is measurable by construction. MSC: 47H10; 13C13; 46A19; 60H25

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