On perfect measures
HTML articles powered by AMS MathViewer
- by G. Koumoullis PDF
- Trans. Amer. Math. Soc. 264 (1981), 521-537 Request permission
Abstract:
Let $\mu$ be a nonzero positive perfect measure on a $\sigma$-algebra of subsets of a set $X$. It is proved that if $\{ {A_i}:i \in I\}$ is a partition of $X$ with ${\mu ^ \ast }({A_i}) = 0$ for all $i \in I$ and the cardinal of $I$ non-(Ulam-) measurable, then there is $J \subset I$ such that ${ \cup _{_{i \in J}}}{A_i}$ is not $\mu$-measurable, generalizing a theorem of Solovay about the Lebesgue measure. This result is used for the study of perfect measures on topological spaces. It is proved that every perfect Borel measure on a metric space is tight if and only if the cardinal of the space is nonmeasurable. The same result is extended to some nonmetric spaces and the relation between perfectness and other smoothness properties of measures on topological spaces is investigated.References
- 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
- N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
- 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
- Richard Haydon, On compactness in spaces of measures and measurecompact spaces, Proc. London Math. Soc. (3) 29 (1974), 1–16. MR 361745, DOI 10.1112/plms/s3-29.1.1
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
- Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
- M. Katětov, Measures in fully normal spaces, Fund. Math. 38 (1951), 73–84. MR 48531, DOI 10.4064/fm-38-1-73-84
- J. H. B. Kemperman and Dorothy Maharam, $R^{c}$ is not almost Lindelöf, Proc. Amer. Math. Soc. 24 (1970), 772–773. MR 253285, DOI 10.1090/S0002-9939-1970-0253285-X
- R. B. Kirk, Measures in topological spaces and $B$-compactness, Nederl. Akad. Wetensch. Proc. Ser. A 72=Indag. Math. 31 (1969), 172–183. MR 0246104 —, Locally compact, $B$-compact spaces, Indag. Math. 31 (1969), 333-344.
- 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
- David J. Lutzer, Another property of the Sorgenfrey line, Compositio Math. 24 (1972), 359–363. MR 307171
- E. Marczewski, On compact measures, Fund. Math. 40 (1953), 113–124. MR 59994, DOI 10.4064/fm-40-1-113-124
- Jan Mařík, The Baire and Borel measure, Czechoslovak Math. J. 7(82) (1957), 248–253 (English, with Russian summary). MR 88532
- E. Michael, $\aleph _{0}$-spaces, J. Math. Mech. 15 (1966), 983–1002. MR 0206907 D. Montgomery, Non-separable metric spaces, Fund. Math. 25 (1935), 527-533.
- W. Moran, The additivity of measures on completely regular spaces, J. London Math. Soc. 43 (1968), 633–639. MR 228645, DOI 10.1112/jlms/s1-43.1.633 —, Measures and mappings on topological spaces, Proc. London Math. Soc. (3) 19 (1969), 493-508.
- W. Moran, Measures on metacompact spaces, Proc. London Math. Soc. (3) 20 (1970), 507–524. MR 437706, DOI 10.1112/plms/s3-20.3.507
- N. Noble and Milton Ulmer, Factoring functions on Cartesian products, Trans. Amer. Math. Soc. 163 (1972), 329–339. MR 288721, DOI 10.1090/S0002-9947-1972-0288721-2 K. Prikry, On images of the Lebesgue measure. I, II, III, manuscripts, 1977.
- K. A. Ross and A. H. Stone, Products of separable spaces, Amer. Math. Monthly 71 (1964), 398–403. MR 164314, DOI 10.2307/2313241
- C. Ryll-Nardzewski, On quasi-compact measures, Fund. Math. 40 (1953), 125–130. MR 59997, DOI 10.4064/fm-40-1-125-130 V. V. Sazonov, On perfect measures, Amer. Math. Soc. Transl. (2) 48 (1965), 229-254.
- Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
- Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961 S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. V. S. Varadarajan, Measures on topological spaces, Amer. Math. Soc. Transl. (2) 48 (1965), 161-228.
- Robert E. Zink, On the structure of measure spaces, Acta Math. 107 (1962), 53–71. MR 140643, DOI 10.1007/BF02545782
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 264 (1981), 521-537
- MSC: Primary 28C15; Secondary 03E55, 54D18, 60A99
- DOI: https://doi.org/10.1090/S0002-9947-1981-0603778-X
- MathSciNet review: 603778