Conditional Analysis on \(\mathbb {R}^d\)

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 \(\sigma \)-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 \(L^0\)-halfspaces. 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 \(L^0\)-convex 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 inf-convolutions.