W. Brannath, W. Schachermayer
Séminaire de Probabilités XXXIII, Springer Lecture Notes in Mathematics, Vol. 1709 (1999), pp. 349-354.
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 $\L$ 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 a ``hereditarily unbounded" part, which seems interesting in its own right.
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