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Cone lattices of upper semicontinuous functions


Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 86 (1982), 81-84
MSC: Primary 26A15; Secondary 41A65, 54B20
DOI: https://doi.org/10.1090/S0002-9939-1982-0663871-9
MathSciNet review: 663871
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Abstract: Let $ X$ be a compact metric space. A well-known theorem of M. H. Stone states that if $ \Omega $ is a vector lattice of continuous functions on $ X$ that separates points and contains a nonzero constant function, then the uniform closure of $ \Omega $ is $ C(X)$. In this article we generalize Stone's sufficient conditions to the upper semicontinuous functions on $ X$ topologized in a natural way.


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

  • [1] G. Beer, A natural topology for upper semicontinuous functions and a Baire category dual for convergence in measure, Pacific J. Math. 96 (1981), 251-263. MR 637972 (83f:54008)
  • [2] -, Upper semicontinuous functions and the Stone Approximation Theorem, J. Approximation Theory 34 (1982), 1-11. MR 647707 (83h:26005)
  • [3] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), 842-850. MR 0448194 (56:6503)
  • [4] S. Dolecki, G. Salinetti and R. Wets, Convergence of functions: equi-semicontinuity, Proc. London Math. Soc. (to appear). MR 684518 (84k:58064)
  • [5] M. H. Stone, A generalized Weierstrass approximation theorem, Studies in Modern Analysis, R. C. Buck, ed., M.A.A. Studies in Math., vol. 1, 1962.
  • [6] R. A. Wijsman, Convergence of sequences of convex sets, cones, and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0663871-9
Keywords: Semicontinuous function, Stone Approximation Theorem, Hausdorff metric, monotone functional
Article copyright: © Copyright 1982 American Mathematical Society

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