# Conditional Analysis on R d

@article{Cheridito2014ConditionalAO, title={Conditional Analysis on R d}, author={Patrick Cheridito and Michael Kupper and Nicolas Vogelpoth}, journal={arXiv: Functional Analysis}, year={2014}, pages={179-211} }

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 �-finite mea- sure space. We study the question when 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, L 0 -half-spaces and L 0 -convex polyhedral sets. We investigate orthogonal complements, orthogonal… Expand

#### 15 Citations

A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem

- Mathematics
- 2015

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the… Expand

On $d$--$\sigma$--stability in random metric spaces and its applications

- Mathematics
- 2019

In 2010, the first author of this paper introduced the notion of σ– stability for a nonempty subset of an L(F , K)–module in [T.X. Guo, Relations between some basic results derived from two kinds of… Expand

Brouwer fixed point theorem in (L0)d

- Mathematics
- 2013

The classical Brouwer fixed point theorem states that in Rd every continuous function from a convex, compact set on itself has a fixed point. For an arbitrary probability space, let L0=L0(Ω,A,P) be… Expand

The algebra of conditional sets and the concepts of conditional topology and compactness

- Mathematics
- 2016

Abstract 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… Expand

On compactness in $L^0$-modules

- Mathematics
- 2017

Several results in functional analysis are extended to the setting of $L^0$-modules, where $L^0$ denotes the ring of all measurable functions $x\colon \Omega\to \mathbb{R}$. The focus is on results… Expand

A perturbational duality approach in vector optimization

- Mathematics
- 2018

A perturbational vector duality approach for objective functions $f\colon X\to \bar{L}^0$ is developed, where $X$ is a Banach space and $\bar{L}^0$ is the space of extended real valued functions on a… Expand

Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time

- Computer Science, Mathematics
- J. Optim. Theory Appl.
- 2020

We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past… Expand

BS\Delta Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

- Mathematics
- 2013

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BSΔEs) and then prove convergence of these to solutions of backward stochastic… Expand

Measures and Integrals in Conditional Set Theory

- Mathematics
- 2017

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by… Expand

Versions of Eberlein-Smulian and Amir-Lindenstrauss theorems in the framework of conditional sets

- Mathematics
- 2015

Based on conditional set theory, we study conditional weak topologies, extending some well-known results to this framework and culminating with the proof of conditional versions of… Expand

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