Abstract not found

. 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 0(\Omega ; F ; P) of real-valued random variables on a probability space (\Omega ; F ; P) 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 L 0 +(\Omega ; F ; P), if we place L 0 +(\Omega ; F ; P) in duality with itself, the scalar product now taking values in [0; 1]. In this setting the order structure of L 0(\Omega ; F ; P) plays an important role and we obtain that the bipolar of a subset of L 0 +(\Omega ; F ; P) equals its closed, convex and solid hull. In the course of the proof we show a decomposition lemma for convex subsets of L 0 +(\Omega ; F ; P) into a "bounded" and a "hereditarily...

Dedicated to the memory of Moez Alimohamed

Motivated by applications in financial mathematics, [3] showed that, although L , P) fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of L , P) : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by [1] in the multidimensional case, replacing by a closed convex cone K of [0, , and by [12], who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of [12] can be extended to the multidimensional case. Using a decomposition result obtained in [3] and [1], we also remove the boundedness assumption of [12] in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones K. Key words: bipolar theorem, convex analysis, partial order.

Motivated by applications in financial mathematics, [3] showed that, although L0 (R+; Ω, F, P) fails to be locally convex, an analogue to the classical bipolar theorem can be obtained for subsets of L0 (R+; Ω, F, P) : if we place this space in polarity with itself, the bipolar of a set of non-negative random variables is equal to its closed (in probability), solid, convex hull. This result was extended by [1] in the multidimensional case, replacing R+ by a closed convex cone K of [0, ∞) d, and by [12], who provided a conditional version in the unidimensional case. In this paper, we show that the conditional bipolar theorem of [12] can be extended to the multidimensional case. Using a decomposition result obtained in [3] and [1], we also remove the boundedness assumption of [12] in the one dimensional case and provide less restrictive assumptions in the multidimensional case. These assumptions are completely removed in the case of polyhedral cones K.