Maximum Lebesgue extension of monotone convex functions
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Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function.
To Appear in Journal of Functional Analysis, 32 pages
Banach lattices, Probability (math.PR), Applications of functional analysis in probability theory and statistics, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Functional Analysis (math.FA), order-continuous Banach lattices, convex risk measures, Mathematics - Functional Analysis, FOS: Economics and business, Risk theory, insurance, Lebesgue property, Risk Management (q-fin.RM), FOS: Mathematics, uniform integrability, order-continuity, Mathematics - Probability, monotone convex functions, Quantitative Finance - Risk Management
Banach lattices, Probability (math.PR), Applications of functional analysis in probability theory and statistics, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), Functional Analysis (math.FA), order-continuous Banach lattices, convex risk measures, Mathematics - Functional Analysis, FOS: Economics and business, Risk theory, insurance, Lebesgue property, Risk Management (q-fin.RM), FOS: Mathematics, uniform integrability, order-continuity, Mathematics - Probability, monotone convex functions, Quantitative Finance - Risk Management
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