Topological spaces whose Baire measure admits a regular Borel extension
HTML articles powered by AMS MathViewer
- by Haruto Ohta and Ken-ichi Tamano PDF
- Trans. Amer. Math. Soc. 317 (1990), 393-415 Request permission
Abstract:
A completely regular, Hausdorff space $X$ is called a Măík space if every Baire measure on $X$ admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence ${F_n} \downarrow \emptyset$ of regular closed sets such that ${ \cap _{n \in \omega }}{B_n} \ne \emptyset$ whenever ${B_n}$’s are Baire sets with ${F_n} \subset {B_n}$; a countably paracompact, non-Măík space; a locally compact, non-Măík space $X$ such that the absolute $E(X)$ is a Măík space; and a locally compact, Măík space $X$ for which $E(X)$ is not. It is also proved that Michael’s product space is not weakly Baire-dominated.References
- W. Adamski, $\tau$-smooth Borel measures on topological spaces, Math. Nachr. 78 (1977), 97–107. MR 492169, DOI 10.1002/mana.19770780108
- Wolfgang Adamski, Extensions of tight set functions with applications in topological measure theory, Trans. Amer. Math. Soc. 283 (1984), no. 1, 353–368. MR 735428, DOI 10.1090/S0002-9947-1984-0735428-9
- George Bachman and Alan Sultan, Measure theoretic techniques in topology and mappings of replete and measure replete spaces, Bull. Austral. Math. Soc. 18 (1978), no. 2, 267–285. MR 499980, DOI 10.1017/S0004972700008078
- George Bachman and Alan Sultan, On regular extensions of measures, Pacific J. Math. 86 (1980), no. 2, 389–395. MR 590550
- W. W. Comfort and S. Negrepontis, Continuous pseudometrics, Lecture Notes in Pure and Applied Mathematics, Vol. 14, Marcel Dekker, Inc., New York, 1975. MR 0410618
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- D. H. Fremlin, Uncountable powers of $\textbf {R}$ can be almost Lindelöf, Manuscripta Math. 22 (1977), no. 1, 77–85. MR 464155, DOI 10.1007/BF01182068
- Zdeněk Frolík, Applications to complete families of continuous functions to the theory of $Q$-spaces, Czechoslovak Math. J. 11(86) (1961), 115–133 (English, with Russian summary). MR 126828
- R. J. Gardner, The regularity of Borel measures and Borel measure-compactness, Proc. London Math. Soc. (3) 30 (1975), 95–113. MR 367145, DOI 10.1112/plms/s3-30.1.95
- R. J. Gardner and W. F. Pfeffer, Borel measures, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 961–1043. MR 776641
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199 S. P. Gul’ko, On properties of subsets of $\Sigma$-products, Soviet Math. Dokl. 18 (1977), 1438-1442.
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- Akio Kato, Union of realcompact spaces and Lindelöf spaces, Canadian J. Math. 31 (1979), no. 6, 1247–1268. MR 553159, DOI 10.4153/CJM-1979-104-8
- J. D. Knowles, Measures on topological spaces, Proc. London Math. Soc. (3) 17 (1967), 139–156. MR 204602, DOI 10.1112/plms/s3-17.1.139
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- John Mack, Directed covers and paracompact spaces, Canadian J. Math. 19 (1967), 649–654. MR 211382, DOI 10.4153/CJM-1967-059-0
- Jan Mařík, The Baire and Borel measure, Czechoslovak Math. J. 7(82) (1957), 248–253 (English, with Russian summary). MR 88532
- S. Mrówka, Some set-theoretic constructions in topology, Fund. Math. 94 (1977), no. 2, 83–92. MR 433388, DOI 10.4064/fm-94-2-83-92 W. Moran, Measures and mappings on topological spaces, Proc. London Math. Soc. 19 (1969), 493-508.
- K. Morita, Paracompactness and product spaces, Fund. Math. 50 (1961/62), 223–236. MR 132525, DOI 10.4064/fm-50-3-223-236
- Susumu Okada and Yoshiaki Okazaki, On measure-compactness and Borel measure-compactness, Osaka Math. J. 15 (1978), no. 1, 183–191. MR 498599
- K. A. Ross and A. H. Stone, Products of separable spaces, Amer. Math. Monthly 71 (1964), 398–403. MR 164314, DOI 10.2307/2313241
- Kenneth A. Ross and Karl Stromberg, Baire sets and Baire measures, Ark. Mat. 6 (1965), 151–160 (1965). MR 196029, DOI 10.1007/BF02591355
- Mary Ellen Rudin, Lectures on set theoretic topology, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 23, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Wyoming, Laramie, Wyo., August 12–16, 1974. MR 0367886
- Milton Ulmer, $C$-embedded $\Sigma$-spaces, Pacific J. Math. 46 (1973), 591–602. MR 380718
- Robert F. Wheeler, Topological measure theory for completely regular spaces and their projective covers, Pacific J. Math. 82 (1979), no. 2, 565–584. MR 551716
- Robert F. Wheeler, Extensions of a $\sigma$-additive measure to the projective cover, Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) Lecture Notes in Math., vol. 794, Springer, Berlin, 1980, pp. 81–104. MR 577963
- Robert F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 1 (1983), no. 2, 97–190. MR 710569
- R. Grant Woods, Ideals of pseudocompact regular closed sets and absolutes of Hewitt realcompactifications, General Topology and Appl. 2 (1972), 315–331. MR 319152
- R. Grant Woods, A survey of absolutes of topological spaces, Topological structures, II (Proc. Sympos. Topology and Geom., Amsterdam, 1978) Math. Centre Tracts, vol. 116, Math. Centrum, Amsterdam, 1979, pp. 323–362. MR 565852
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 393-415
- MSC: Primary 28C15; Secondary 54C50, 54G20
- DOI: https://doi.org/10.1090/S0002-9947-1990-0946425-5
- MathSciNet review: 946425