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.
Similar content being viewed by others
Notes
Regardless of \(\varPhi \in \varDelta _2\) and convexity, \(\sigma (L_{\varPhi ^*},L_\varPhi )\)-closed \(\Rightarrow \) order closed \(\Rightarrow \) norm closed since \(L_\varPhi \) is identified with the order continuous dual of\(L_{\varPhi ^*}\) and norm convergent sequences have order convergent subsequences; see e.g. [22, Chapter 14] for details and unexplained terminologies.
References
Albiac, F.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233, 2nd edn. Springer, Berlin (2016)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics. Mathematical Surveys and Monographs, vol. 105, 2nd edn. American Mathematical Society, Providence (2003)
Andô, T.: Linear functionals on Orlicz spaces. Nieuw Arch. Wisk. 3(8), 1–16 (1960)
Banach, S.: Théorie des opérations linéaires, Monografie Matematyczne, vol. 1. Instytut Matematyczny Polskiej Akademi Nauk, Warszawa. Reprinted by Chelsea Publishing Co., 1955 (1932)
Biagini, S., Frittelli, M.: On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures. In: Delbaen, F., Rásonyi, M., Stricker, C. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance, pp. 1–28. Springer, Berlin (2009)
Delbaen, F.: Differentiability properties of utility functions. In: Delbaen, F., Rásonyi, M., Stricker, C. (eds.) Optimality and Risk—Modern Trends in Mathematical Finance, pp. 39–48. Springer, Berlin (2009)
Delbaen, F.: Monetary Utility Functions. Osaka University CSFI Lecture Notes Series, vol. 3. Osaka University Press (2012)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter & Co., Berlin (2011)
Gao, N., Leung, D.H., Xanthos, F.: The dual representation problem of risk measures (2016). arXiv:1610.08806
Gao, N., Xanthos, F.: On the \(C\)-property and \(w^*\)-representations of risk measures. Math. Finance 28(2), 748–754 (2018)
Grothendieck, A.: Topological vector spaces. Gordon and Breach Science Publishers, New York. Translated from the French by Orlando Chaljub, Notes on Mathematics and its Applications (1973)
Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. In: Kusuoka, S., Maruyama, T. (eds.) Advances in Mathematical Economics, vol. 9, pp. 49–71. Springer, Tokyo (2006)
Komlós, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hung. 18, 217–229 (1967)
Lindenstrauss, J.: Classical Banach Spaces. II. Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97. Springer, Berlin (1979)
Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)
Moreau, J.-J.: Sur la fonction polaire d’une fonction semi-continue supérieurement. C. R. Acad. Sci. Paris 258, 1128–1130 (1964)
Nowak, M.: On the order structure of Orlicz lattices (1989). Bull. Polish Acad. Sci. Math. 36, 239–249 (1988)
Ostrovskii, M.I.: Weak* sequential closures in Banach space theory and their applications. In: Banakh, T. (ed.) General Topology in Banach Spaces, pp. 21–34. Nova Science Publishers, Huntington (2001)
Ostrovskii, M.I.: Weak* closures and derived sets in dual Banach spaces. Note Mat. 31, 129–138 (2011)
Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker Inc., New York (1991)
Zaanen, A.C.: Riesz spaces. II. North-Holland Mathematical Library, vol. 30. North-Holland Publishing Co., Amsterdam (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Freddy Delbaen: Part of this work was done while the first named author was on visit at Tokyo Metropolitan University. Keita Owari: Supported in part by JSPS Grant Number JP17K14210.
Appendix
Appendix
Proof of Proposition 1.2
Only (3) \(\Rightarrow \) (1) deserves a proof. (3) implies, by Proposition 1.1, \(f=f^{**}\), and \(|\mathbb {E}[\eta {\mathbb {1}}_A]|\le \frac{1}{n}\left( f(n{\mathbb {1}}_A)\vee f(-n{\mathbb {1}}_A)+c\right) \) for \(A\in \mathcal {F}\) and \(\eta \in L_1\) with \(f^*(\eta )\le c\) by Young’s inequality; thus (3) implies that \(\{\eta \in L_1: f^*(\eta )\le c\}\) is uniformly integrable, hence \(\sigma (L_1,L_\infty )\)-compact by the Dunford-Pettis theorem. Now Moreau’s theorem [17] shows that f is \(\tau (L_\infty ,L_1)\)-continuous. \(\square \)
Proof of Lemma 2.1
For each \(\xi \in L_{\varPhi ^*}\), \(\eta \mapsto \eta \xi \) continuously maps \((L_\varPhi ,\sigma (L_\varPhi ,L_{\varPhi ^*}))\) into \((L_1,\sigma (L_1,L_\infty ))\) since \(\xi \zeta \in L_1\), \(\forall \zeta \in L_\infty \). Thus if A is relatively \(\sigma (L_\varPhi ,L_{\varPhi ^*})\)-compact, its image \(A\xi \) is relatively weakly compact in \(L_1\), i.e. uniformly integrable. Conversely, if \(A\xi \), \(\xi \in L_{\varPhi ^*}\), are uniformly integrable, then \(c_\xi :=\sup _{\eta \in A}\mathbb {E}[|\eta \xi |]<\infty \) for each \(\xi \in L_{\varPhi ^*}\), so A is pointwise bounded in the algebraic dual \(L_{\varPhi ^*}^\#\) of \(L_{\varPhi ^*}\), and A is relatively \(\sigma (L_1,L_\infty )\)-compact in \(L_1\). Thus if \((\eta _\alpha )_\alpha \) is a net in A with the pointwise limit \(f(\xi )=\lim _\alpha \mathbb {E}[\eta _\alpha \xi ]\) in \(L_{\varPhi ^*}^\#\), there is a unique \(\eta _0\in L_1\) such that \(f|_{L_\infty }(\xi )=\mathbb {E}[\eta _0\xi ]\) for \(\xi \in L_\infty \). Then for each \(\xi \in L_{\varPhi ^*}\), \(\mathbb {E}[|\eta _0\xi |]=\sup _n\mathbb {E}[\eta _0\xi {\mathbb {1}}_{\{|\xi |\le n\}} \text {sgn}(\eta _0\xi )] =\sup _nf(\xi {\mathbb {1}}_{\{|\xi |\le n\}}\text {sgn}(\eta _0\xi )) \le c_\xi \), hence \(\eta _0\in L_\varPhi \), while \(|f(\xi )-f(\xi {\mathbb {1}}_{\{|\xi |\le n\}})|=|f(\xi {\mathbb {1}}_{\{|\xi |>n\}})|\le \sup _{\eta \in A}\mathbb {E}[|\eta \xi |{\mathbb {1}}_{\{|\xi |>n\}}] \rightarrow 0\) since \(A\xi \) is uniformly integrable; hence \(f(\xi )=\mathbb {E}[\eta _0\xi ]\). Therefore A is pointwise bounded and its \(\sigma (L_{\varPhi ^*}^\#,L_{\varPhi ^*})\)-closure in \(L_{\varPhi ^*}^\#\) lies in \(L_\varPhi \); hence A is relatively \(\sigma (L_\varPhi ,L_{\varPhi ^*})\)-compact.\(\square \)
Rights and permissions
About this article
Cite this article
Delbaen, F., Owari, K. Convex functions on dual Orlicz spaces. Positivity 23, 1051–1064 (2019). https://doi.org/10.1007/s11117-019-00651-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-019-00651-x