A bipolar theorem for

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 space equals its closed convex hull.

The space of real-valued random variables on a probability space 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 , if we place in duality with itself, the scalar product now taking values in [0, ∞]. In this setting the order structure of plays an important role and we obtain that the bipolar of a subset of equals its closed, convex and solid hull.

In the course of the proof we show a decomposition lemma for convex subsets of $L_{^ + }^0 \left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ into a “bounded” and a “hereditarily unbounded” part, which seems interesting in its own right.

The research of this paper was financially supported by the Austrian Science Foundation (FWF) under grant SFB#10 (‘Adaptive Information Systems and Modelling in Economics and Management Science’)