This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring L 0 of measurable functions on a σ-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of L 0 -affine sets, L 0 -convex sets, L 0 -convex cones, L 0 -hyperplanes and L 0halfspaces. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L 0 -linear, L 0 -affine, L 0 -convex and L 0 -sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano-Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of L 0 -convex sets by L 0 -hyperplanes and study L 0convex conjugate functions. We provide a result on the existence of L 0 -subgradients of L 0 -convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional infconvolutions. § We thank Ramon van Handel, Ying Hu, Asgar Jamneshan, Mitja Stadje and Martin Streckfuß for fruitful discussions and helpful comments.Unless Ω is the union of finitely many atoms, (L 0 ) d is an infinite-dimensional vector space over R. But conditioned on F , it is only d-dimensional. Or put differently, it is a free module of rank d over the ring L 0 . This allows us to derive conditional analogs of classical results from linear algebra, real analysis and convex analysis that depend on the fact that R d is a finite-dimensional vector space. L 0 -modules have been studied before; see, for instance, Filipović et al. (2009), Kupper and, Guo (2010), Guo (2011) and the references in these papers. But since we consider free modules of finite rank, we are able to provide stronger results under weaker assumptions, and moreover, do not need Zorn's lemma or the axiom of choice. Our approach differs from standard measurable selection arguments in that we work modulo null-sets with respect to the measure µ and do not use ω-wise arguments. This has the advantage that one never leaves the world of measurable functions. But it only works in situations where a measure µ is given, and the quantities of interest do not depend on µ-null sets.The results in this paper are theoretical. But they have already been applied several times: in Cheridito and Hu (2011), they were used to describe stochastic constraints and characterize optimal strategies in a dynamic consumption and investment problem. In Cheridito and Stadje (2012) they guaranteed the existence of a conditional subgradient. In Cheridito et al. (2012) they were applied to show existence and uniqueness of economic equilibria in incomplete market models.The structure of the paper is as follows: In Section 2 we investigate when an L 0 -submodule of (L 0 ) d is finitely generated. Then we study conditional o...
