Borel measures and Hausdorff distance

Beer, Gerald; Villar, Luzviminda

doi:10.1090/S0002-9947-1988-0940226-0

Borel measures and Hausdorff distance
HTML articles powered by AMS MathViewer

by Gerald Beer and Luzviminda Villar PDF

Trans. Amer. Math. Soc. 307 (1988), 763-772 Request permission

Abstract:

In this article we study the restriction of Borel measures defined on a metric space $X$ to the nonempty closed subsets $\operatorname {CL} (X)$ of $X$, topologized by Hausdorff distance. We show that a $\sigma$-finite Radon measure is a Borel function on $\operatorname {CL} (X)$, and characterize those $X$ for which each outer regular Radon measure on $X$ is semicontinuous when restricted to $\operatorname {CL} (X)$. A number of density theorems for Radon measures are also presented.

References

Masahiko Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958), 11–16; erratum, 941. MR 99023
Gerald Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95 (1985), no. 4, 653–658. MR 810180, DOI 10.1090/S0002-9939-1985-0810180-3
Gerald Beer and Luzviminda Villar, When is a free union of regular measures regular?, Amer. Math. Monthly 94 (1987), no. 5, 443–445. MR 886306, DOI 10.2307/2322729
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
R. J. Gardner, The regularity of Borel measures, Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) Lecture Notes in Math., vol. 945, Springer, Berlin-New York, 1982, pp. 42–100. MR 675272
R. J. Gardner and W. F. Pfeffer, Borel measures, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 961–1043. MR 776641
Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
M. R. Krom, Cartesian products of metric Baire spaces, Proc. Amer. Math. Soc. 42 (1974), 588–594. MR 334138, DOI 10.1090/S0002-9939-1974-0334138-9
Robert A. McCoy, Baire spaces and hyperspaces, Pacific J. Math. 58 (1975), no. 1, 133–142. MR 410689
Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
Sam B. Nadler Jr. and Thelma West, A note on Lebesgue spaces, Topology Proc. 6 (1981), no. 2, 363–369 (1982). MR 672468
K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
Washek F. Pfeffer, Integrals and measures, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 42, Marcel Dekker, Inc., New York-Basel, 1977. MR 0460580
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
W. C. Waterhouse, On $UC$ spaces, Amer. Math. Monthly 72 (1965), 634–635. MR 184200, DOI 10.2307/2313854

Elementary problem

93