TY - UNPB

T1 - A bipolar theorem for $L^0_+(\Om, \Cal F, \P)$

AU - Brannath, Werner

AU - Schachermayer, Walter

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

U2 - 10.57938/67a2eb1f-f061-4809-a5d3-d2b6aeda8a54

DO - 10.57938/67a2eb1f-f061-4809-a5d3-d2b6aeda8a54

M3 - WU Working Paper

T3 - Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

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

PB - SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business

CY - Vienna

ER -