A bipolar theorem for $L^0_+(\Om, \Cal F, \P)$

Werner Brannath, Walter Schachermayer

Publikation: Working/Discussion PaperWU Working Paper

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Abstract

A consequence of the Hahn-Banach 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 real-valued 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.

Publikationsreihe

ReiheWorking Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"
Nummer28

WU Working Paper Reihe

  • Working Papers SFB \Adaptive Information Systems and Modelling in Economics and Management Science\

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