Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A Stone-Weierstrass theorem without closure under suprema
HTML articles powered by AMS MathViewer

by R. Preston McAfee and Philip J. Reny PDF
Proc. Amer. Math. Soc. 114 (1992), 61-67 Request permission

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
  • Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
  • 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
  • Avner Friedman, Foundations of modern analysis, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100
  • 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, DOI 10.2307/1913717
  • 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, DOI 10.1111/1468-0262.00473
  • H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Similar Articles
Additional Information
  • © Copyright 1992 American Mathematical Society
  • 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