Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A Stone-Weierstrass theorem without closure under suprema


Authors: R. Preston McAfee and Philip J. Reny
Journal: Proc. Amer. Math. Soc. 114 (1992), 61-67
MSC: Primary 41A65; Secondary 41A10, 46E25, 54C40
DOI: https://doi.org/10.1090/S0002-9939-1992-1091186-2
MathSciNet review: 1091186
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a compact metric space $ X$, consider a linear subspace $ A$ of $ C\left( X \right)$ containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if $ A$ separates points, then the closure of $ A$ under both minima and maxima is dense in $ C\left( X \right)$. By the Hahn-Banach Theorem, if $ A$ separates probability measures, $ A$ is dense in $ C\left( X \right)$. It is shown that if $ A$ separates points from probability measures, then the closure of $ A$ under minima is dense in $ C\left( X \right)$. This theorem has applications in economic theory.


References [Enhancements On Off] (What's this?)

  • [1] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
  • [2] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
    Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1009163
    Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. MR 1009164
  • [3] Avner Friedman, Foundations of modern analysis, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100
  • [4] R. Preston McAfee, John McMillan, and Philip J. Reny, Extracting the surplus in the common-value auction, Econometrica 57 (1989), no. 6, 1451–1459. MR 1035121, https://doi.org/10.2307/1913717
  • [5] Erzo G. J. Luttmer and Thomas Mariotti, The existence of subgame-perfect equilibrium in continuous games with almost perfect information: a comment on “The existence of subgame-perfect equilibrium in continuous games with almost perfect information: a case for public randomization” [Econometrica 63 (1995), no. 3, 507–544; MR1334862] by C. Harris, P. Reny and A. Robson, Econometrica 71 (2003), no. 6, 1909–1911. MR 2015423, https://doi.org/10.1111/1468-0262.00473
  • [6] H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR 0151555

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A65, 41A10, 46E25, 54C40

Retrieve articles in all journals with MSC: 41A65, 41A10, 46E25, 54C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1091186-2
Article copyright: © Copyright 1992 American Mathematical Society