A bipolar theorem for Open image in new window

  • W. Brannath
  • W. Schachermayer
Autres Exposés
Part of the Lecture Notes in Mathematics book series (LNM, volume 1709)


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 Open image in new window of real-valued random variables on a probability space Open image in new window 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 Open image in new window , if we place Open image in new window in duality with itself, the scalar product now taking values in [0, ∞]. In this setting the order structure of Open image in new window plays an important role and we obtain that the bipolar of a subset of Open image in new window 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.

1980 Mathematics Subject Classification (1991 Revision)

Primary: 62B20 28A99 26A20 52A05 46A55 Secondary: 46A40 46N10 90A09 Key words and phrases Convex sets of measurable functions Bipolar theorem bounded in probability hereditarily unbounded 


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© Springer-Verlag 1999

Authors and Affiliations

  • W. Brannath
  • W. Schachermayer

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