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The Shapley-Folkman Theorem and the Range of a Bounded Measure: An Elementary and Unified Treatment


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  • M. Ali Khan
  • Kali P. Rath


We present proofs, based on the Shapley-Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov's celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley-Folkman theorem in mathematical economics.

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Paper provided by The Johns Hopkins University,Department of Economics in its series Economics Working Paper Archive with number 586.

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Date of creation: Dec 2011
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Handle: RePEc:jhu:papers:586

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  1. Carmona, Guilherme & Podczeck, Konrad, 2009. "On the existence of pure-strategy equilibria in large games," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1300-1319, May.
  2. Carmona, Guilherme, 2003. "On the Purification of Nash Equilibria of Large Games," FEUNL Working Paper Series wp436, Universidade Nova de Lisboa, Faculdade de Economia.
  3. Zhou, Lin, 1993. "A Simple Proof of the Shapley-Folkman Theorem," Economic Theory, Springer, vol. 3(2), pages 371-72, April.
  4. Anderson, Robert M. & Zame, William R., 1995. "Edgeworth's Conjecture with Infinitely Many Commodities," Department of Economics, Working Paper Series qt5kb2x3cd, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  5. Henry, Claude, 1972. "Market Games with Indivisible Commodities and Non-convex Preferences," Review of Economic Studies, Wiley Blackwell, vol. 39(1), pages 73-76, January.
  6. Starr, Ross M., 1981. "Approximation of points of the convex hull of a sum of sets by points of the sum: An elementary approach," Journal of Economic Theory, Elsevier, vol. 25(2), pages 314-317, October.
  7. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  8. Robert M. Anderson & M. Ali Khan & Salim Rashid, 1981. "Approximate Equilibria with Bounds Independent of Preferences," Cowles Foundation Discussion Papers 595, Cowles Foundation for Research in Economics, Yale University.
  9. Heller, Walter Perrin, 1972. "Transactions with set-up costs," Journal of Economic Theory, Elsevier, vol. 4(3), pages 465-478, June.
  10. Anderson, Robert M, 1982. "A Market Value Approach to Approximate Equilibria," Econometrica, Econometric Society, vol. 50(1), pages 127-36, January.
  11. Carmona, Guilherme, 2008. "Purification of Bayesian-Nash equilibria in large games with compact type and action spaces," Journal of Mathematical Economics, Elsevier, vol. 44(12), pages 1302-1311, December.
  12. Anderson, Robert M., 1987. "Gap-minimizing prices and quadratic core convergence," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 1-15, February.
  13. Broome, John, 1972. "Approximate equilibrium in economies with indivisible commodities," Journal of Economic Theory, Elsevier, vol. 5(2), pages 224-249, October.
  14. Geller, William, 1986. "An Improved Bound for Approximate Equilibria [Approximate Equilibria with Bounds Independent of Preferences]," Review of Economic Studies, Wiley Blackwell, vol. 53(2), pages 307-08, April.
  15. Shaked, A., 1976. "Absolute approximations to equilibrium in markets with non-convex preferences," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 185-196, July.
  16. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-87, November.
  17. Yannelis, Nicholas C., 1983. "Existence and fairness of value allocation without convex preferences," Journal of Economic Theory, Elsevier, vol. 31(2), pages 283-292, December.
  18. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
  19. Anderson, Robert M, 1988. "The Second Welfare Theorem with Nonconvex Preferences," Econometrica, Econometric Society, vol. 56(2), pages 361-82, March.
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