Cone lattices of upper semicontinuous functions

Author:
Gerald Beer

Journal:
Proc. Amer. Math. Soc. **86** (1982), 81-84

MSC:
Primary 26A15; Secondary 41A65, 54B20

MathSciNet review:
663871

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Abstract: Let be a compact metric space. A well-known theorem of M. H. Stone states that if is a vector lattice of continuous functions on that separates points and contains a nonzero constant function, then the uniform closure of is . In this article we generalize Stone's sufficient conditions to the upper semicontinuous functions on topologized in a natural way.

**[1]**Gerald Beer,*A natural topology for upper semicontinuous functions and a Baire category dual for convergence in measure*, Pacific J. Math.**96**(1981), no. 2, 251–263. MR**637972****[2]**Gerald Beer,*Upper semicontinuous functions and the Stone approximation theorem*, J. Approx. Theory**34**(1982), no. 1, 1–11. MR**647707**, 10.1016/0021-9045(82)90110-1**[3]**Ennio De Giorgi and Tullio Franzoni,*Su un tipo di convergenza variazionale*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**58**(1975), no. 6, 842–850 (Italian). MR**0448194****[4]**Szymon Dolecki, Gabriella Salinetti, and Roger J.-B. Wets,*Convergence of functions: equi-semicontinuity*, Trans. Amer. Math. Soc.**276**(1983), no. 1, 409–430. MR**684518**, 10.1090/S0002-9947-1983-0684518-7**[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**, 10.1090/S0002-9947-1966-0196599-8

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Additional Information

DOI:
http://dx.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