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Positivity

, Volume 20, Issue 2, pp 343–354 | Cite as

An exact Fatou lemma for Gelfand integrals: a characterization of the Fatou property

  • M. Ali Khan
  • Nobusumi SagaraEmail author
  • Takashi Suzuki
Article

Abstract

We establish an exact version of Fatou’s lemma for Gelfand integrals of functions and multifunctions in dual Banach spaces without any order structure, and under the saturation property on the underlying measure space. The necessity and sufficiency of saturation for the Fatou property is demonstrated. Our result has a direct application to the equilibrium existence result for saturated economies without convexity assumptions.

Keywords

Gelfand integral Fatou’s lemma Saturated measure space \(\mathrm{Weak}^*\) upper-closure property 

Mathematics Subject Classification

Primary 28B05 28B20 Secondary 46G10 

Notes

Acknowledgments

This paper was presented at the “Summer Workshop in Economic Theory (SWET)” at the Centre d’Economie de la Sorbonne (CES), Université Paris 1 Panthéon-Sorbonne, in honor of the 65th birthday of Professor Bernard Cornet, on June 26–28, 2014, at the “Annual Meeting of the Mathematical Society of Japan” at Meiji University on March 21–24, 2015, and at the “11th Annual Conference on General Equilibrium and its Applications and a Celebration of Donald Brown” at Cowles Foundation for Research in Economics, Yale University, on April 24–26, 2015. The authors acknowledge helpful comments of Erik Balder, Don Brown, Jun Kawabe, Boris Mordukhovich, Konrad Podczeck, and the careful reading of an anonymous referee. This research is supported by Grant-in-Aids for Scientific Research (No. 26380246 and No. 15K03362) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of EconomicsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Department of EconomicsHosei UniversityMachidaJapan
  3. 3.Department of EconomicsMeiji-Gakuin UniversityMinatoJapan

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