Brannath, Werner and Schachermayer, Walter (1999) A bipolar theorem for $L^0_+(\Om, \Cal F, \P)$. Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science", 28. SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, Vienna.

PDF
Download (149Kb)  Preview 
Abstract
A consequence of the HahnBanach theorem is the classical bipolar theorem which states that the bipolar of a subset of a locally convex vector pace equals its closed convex hull. The space $\L$ of realvalued random variables on a probability space $\OF$ equipped with the topology of convergence in measure fails to be locally convex so that  a priori  the classical bipolar theorem does not apply. In this note we show an analogue of the bipolar theorem for subsets of the positive orthant $\LO$, if we place $\LO$ in duality with itself, the scalar product now taking values in $[0, \infty]$. In this setting the order structure of $\L$ plays an important role and we obtain that the bipolar of a subset of $\LO$ equals its closed, convex and solid hull. In the course of the proof we show a decomposition lemma for convex subsets of $\LO$ into a "bounded" and "hereditarily unbounded" part, which seems interesting in its own right. (author's abstract)
Item Type:  Paper 

Keywords:  convex sets of measurable functions / bipolar theorem / bounded in probability / hereditarily unbounded. 
Divisions:  Departments > Informationsverarbeitung u Prozessmanag. > Produktionsmanagement > Taudes Departments > Finance, Accounting and Statistics > Statistics and Mathematics Departments > Marketing > Service Marketing und Tourismus Departments > Informationsverarbeitung u Prozessmanag. > Informationswirtschaft 
Depositing User:  Repository Administrator 
Date Deposited:  22 Mar 2002 13:14 
Last Modified:  28 Mar 2015 03:37 
URI:  http://epub.wu.ac.at/id/eprint/1688 
Actions
View Item 