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Proceedings of the American Mathematical Society
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
MathSciNet review: 1091186
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1091186-2
PII: S 0002-9939(1992)1091186-2
Article copyright: © Copyright 1992 American Mathematical Society