, 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


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.


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

Mathematics Subject Classification

Primary 28B05 28B20 Secondary 46G10 



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.


  1. 1.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Aumann, R.J.: Markets with a continum of traders. Econometrica 32, 39–50 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aumann, R.J.: Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balder, E.J.: Fatou’s lemma for Gelfand integrals by means of Young measure theory. Positivity 6, 317–329 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Balder, E.J., Sambucini, A.R.: Fatou’s lemma for multifunctions with unbounded values in a dual space. J. Convex Anal. 12, 383–395 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bewley, T.F.: The equality of the core and the set of equilibria in economies with infinitely many commodities and a continuum of agents. Internat. Econ. Rev. 14, 383–394 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bewley, T.F.: A very weak theorem on the existence of equilibria in atomless economies with infinitely many commodities. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, pp. 224–232. Springer, Berlin (1991)CrossRefGoogle Scholar
  8. 8.
    Castaing, C., Saadoune, M.: Convergence in a dual space with applications to Fatou lemma. Adv. Math. Econ. 12, 23–69 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math, vol. 580. Springer, Berlin (1977)CrossRefzbMATHGoogle Scholar
  10. 10.
    Cornet, B., Medecin, J.-P.: Fatou’s lemma for Gelfand integrable mappings. Positivity 6, 297–315 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diestel, J., Uhl Jr, J.J.: Vector Measures. Am. Math. Soc, Providence, RI (1977)Google Scholar
  12. 12.
    Fajardo, S., Keisler, H.J.: Model Theory of Stochastic Processes. A K Peters Ltd, Natick (2002)zbMATHGoogle Scholar
  13. 13.
    Fremlin, D.H.: Measure Theory, vol. 3, Parts I and II: Measure Algebras, 2nd edn. Torres Fremlin, Colchester (2012)Google Scholar
  14. 14.
    Hildenbrand, W.: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton (1974)zbMATHGoogle Scholar
  15. 15.
    Hoover, D., Keisler, H.J.: Adapted probability distributions. Trans. Am. Math. Soc. 286, 159–201 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Keisler, H.J., Sun, Y.N.: Why saturated probability spaces are necessary. Adv. Math. 221, 1584–1607 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khan, M.A.: On the integration of set-valued mappings in a non-reflexive Banach space, II. Simon Stevin 59, 257–267 (1985)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Khan, M.A.: Equilibrium points of nonatomic games over a Banach spaces. Trans. Am. Math. Soc. 293, 737–749 (1986)CrossRefzbMATHGoogle Scholar
  19. 19.
    Khan, M.A., Majumdar, M.K.: Weak sequential convergence in \(L_1(\mu, X)\) and an approximate version of Fatou’s lemma. J. Math. Anal. Appl. 114, 569–573 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Khan, M.A., Sagara, N.: Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Ill. J. Math. 57, 145–169 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Khan, M.A., Sagara, N.: Weak sequential convergence in \(L^1(\mu, X)\) and an exact version of Fatou’s lemma. J. Math. Anal. Appl. 412, 554–563 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khan, M.A., Sagara, N.: Maharam-types and Lyapunov’s theorem for vector measures on locally convex spaces with control measures. J. Convex Anal. 22, (2015) (in press)Google Scholar
  23. 23.
    Loeb, P.A., Sun, Y.N.: A general Fatou lemma. Adv. Math. 213, 741–762 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Loeb, P.A., Sun, Y.N.: Purification and saturation. Proc. Am. Math. Soc. 137, 2719–2724 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Maharam, D.: On homogeneous measure algebras. Proc. Natl. Acad. Sci. USA 28, 108–111 (1942)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ostroy, J.M., Zame, W.: Nonatomic economies and the boundaries of perfect competition. Econometrica 62, 593–633 (1994)CrossRefzbMATHGoogle Scholar
  27. 27.
    Podczeck, K.: Markets with infinitely many commodities and a continuum of agents with non-convex preferences. Econ. Theory 9, 385–426 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Podczeck, K.: On the convexity and compactness of the integral of a Banach space valued correspondence. J. Math. Econ. 44, 836–852 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rustichini, A.: A counterexample and an exact version of Fatou’s lemma in infinite dimensional spaces. Arch. Math. 52, 357–362 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sagara, N., Suzuki, T.: Exchange economies with infinitely many commodities and nonconvex preferences. Dept. Econ. Meiji-Gakuin Univ., Mimeo (2014)Google Scholar
  31. 31.
    Schmeidler, D.: Fatou’s lemma in several dimensions. Proc. Am. Math. Soc. 24, 300–306 (1970)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7, 295–300 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sun, Y.N.: Integration of correspondences on Loeb spaces. Trans. Am. Math. Soc. 349, 129–153 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sun, Y.N., Yannelis, N.C.: Saturation and the integration of Banach valued correspondences. J. Math. Econ. 44, 861–865 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Suzuki, T.: Competitive equilibria of a large economy on the commodity space \(\ell ^\infty \). Adv. Math. Econ. 17, 1–19 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Suzuki, T.: Core and competitive equilibria for a coalitional exchange economy with infinite time horizon. J. Math. Econ. 49, 234–244 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Suzuki, T.: An elementary proof of an infinite dimensional Fatou’s lemma with an application to market equilibrium analysis. J. Pure Appl. Math. 10, 159–182 (2013)Google Scholar
  38. 38.
    Thomas, G.E.F.: Integration of functions with values in locally convex Suslin spaces. Trans. Am. Math. Soc. 212, 61–81 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Uhl Jr, J.J.: The range of a vector-valued measure. Proc. Am. Math. Soc. 23, 158–163 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yannelis, N.C.: Integration of Banach-valued correspondences. In: Khan, M.A., Yannelis, N.C. (eds.) Equilibrium Theory in Infinite Dimensional Spaces, pp. 2–35. Springer, Berlin (1991)CrossRefGoogle Scholar

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© 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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