Conditional Analysis on ℝ𝑑

Abstract

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 𝐿0 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 𝐿0 -affine sets, 𝐿0 -convex sets, 𝐿0 -convex cones, 𝐿0 -hyperplanes and 𝐿0 -halfspaces. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study 𝐿0 -linear, 𝐿0 -affine, 𝐿0 -convex and 𝐿0 -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 𝐿0 -convex sets by 𝐿0 -hyperplanes and study 𝐿0 -convex conjugate functions. We provide a result on the existence of 𝐿0 -subgradients of 𝐿0 -convex functions, prove a conditional version of the Fenchel–Moreau theorem and study conditional inf-convolutions.