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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)

##### MSC:
 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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