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Positivity

pp 1–14 | Cite as

Convex functions on dual Orlicz spaces

  • Freddy Delbaen
  • Keita OwariEmail author
Article
  • 21 Downloads

Abstract

In the dual \(L_{\varPhi ^*}\) of a \(\varDelta _2\)-Orlicz space \(L_\varPhi \), that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology \(\tau (L_{\varPhi ^*},L_\varPhi )\) if and only if on each order interval \([-\zeta ,\zeta ]=\{\xi : -\zeta \le \xi \le \zeta \}\) (\(\zeta \in L_{\varPhi ^*}\)), it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence \((\xi _n)_n\) in \(L_{\varPhi ^*}\) admits a sequence of forward convex combinations \({{\bar{\xi }}}_n\in \text {conv}(\xi _n,\xi _{n+1},\ldots )\) such that \(\sup _n|{\bar{\xi }}_n|\in L_{\varPhi ^*}\) and \({\bar{\xi }}_n\) converges a.s.

Keywords

Orlicz spaces Mackey topology Komlós’s theorem Convex functions Order closed sets Risk measures 

Mathematics Subject Classification

46E30 46A55 52A41 46B09 91G80 46B10 91B30 

Notes

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Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZürichSwitzerland
  2. 2.Institute of MathematicsUniversity of ZürichZürichSwitzerland
  3. 3.Department of Mathematical SciencesRitsumeikan UniversityKusatsuJapan

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