IDEAS home Printed from
MyIDEAS: Login to save this paper or follow this series

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

  • 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
Download Restriction: no

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

in new window

Date of creation: Dec 2011
Date of revision:
Handle: RePEc:jhu:papers:586
Contact details of provider: Postal: 3400 North Charles Street Baltimore, MD 21218
Phone: 410-516-7601
Fax: 410-516-7600
Web page:

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
  1. Rashid, Salim, 1983. "Equilibrium points of non-atomic games : Asymptotic results," Economics Letters, Elsevier, vol. 12(1), pages 7-10.
  2. Anderson, Robert M., 1987. "Gap-minimizing prices and quadratic core convergence," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 1-15, February.
  3. Robert M. Anderson and William R. Zame., 1995. "Edgeworth's Conjecture with Infinitely Many Commodities," Economics Working Papers 95-235, University of California at Berkeley.
  4. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-87, November.
  5. Anderson, Robert M, 1988. "The Second Welfare Theorem with Nonconvex Preferences," Econometrica, Econometric Society, vol. 56(2), pages 361-82, March.
  6. 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.
  7. Zhou, Lin, 1993. "A Simple Proof of the Shapley-Folkman Theorem," Economic Theory, Springer, vol. 3(2), pages 371-72, April.
  8. 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.
  9. 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.
  10. Carmona, Guilherme & Podczeckz, Konrad, 2008. "On the Existence of Pure-Strategy Equilibria in Large Games," FEUNL Working Paper Series wp531, Universidade Nova de Lisboa, Faculdade de Economia.
  11. Carmona, Guilherme, 2003. "On the Purification of Nash Equilibria of Large Games," FEUNL Working Paper Series wp436, Universidade Nova de Lisboa, Faculdade de Economia.
  12. Broome, John, 1972. "Approximate equilibrium in economies with indivisible commodities," Journal of Economic Theory, Elsevier, vol. 5(2), pages 224-249, October.
  13. Anderson, Robert M, 1982. "A Market Value Approach to Approximate Equilibria," Econometrica, Econometric Society, vol. 50(1), pages 127-36, January.
  14. 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.
  15. Heller, Walter Perrin, 1972. "Transactions with set-up costs," Journal of Economic Theory, Elsevier, vol. 4(3), pages 465-478, June.
  16. 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.
  17. 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.
  18. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
  19. 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.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:jhu:papers:586. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (None)

The email address of this maintainer does not seem to be valid anymore. Please ask None to update the entry or send us the correct address

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.