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A bipolar theorem for \(L_+^0(\Omega,{\mathcal F},\mathbb{P})\). (English) Zbl 0957.46020
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 349-354 (1999).
Let \(L^0:=L^0(\Omega,{\mathcal F},{\mathbb P})\) be the space of real-valued random variables on \((\Omega,{\mathcal F},{\mathbb P})\) equipped with the topology of convergence in probability. Because of the failure of local convexity, the classical bipolar theorem is not available on \(L^0\). On \(L_+^0\), the orthant of positive elements in \(L^0\), the notion of a polar \(C^0\) of a set \(C\subseteq L_+^0\) can be defined by \[ C^0=\{g\in L_+^0: {\mathbb E}[fg]\leq 1,\text{ for each }f\in C\Big\}. \] The main result can now be stated as follows:
For \(C\subseteq L_+^0\), the bipolar \(C^{00}\) is the smallest, closed, convex, solid set in \(L_+^0\) containing \(C\).
The proof relies on an interesting decomposition on \(\Omega\) into two sets \(\Omega_u\) and \(\Omega_b\) such that \(C\big|_ {\Omega_b}\) is bounded in probability and \(C\) is \`\` hereditarily unbounded in probability” on \(\Omega_u\).
For the entire collection see [Zbl 0924.00016].
Reviewer: A.Schied (Berlin)

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
46A55 Convex sets in topological linear spaces; Choquet theory
91B70 Stochastic models in economics
52A05 Convex sets without dimension restrictions (aspects of convex geometry)
46A20 Duality theory for topological vector spaces
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