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Abstrakty
We present arguments showing that the standard notion of the support of a probabilistic Borel measure is not well defined in every topological space. Our goal is to create a "very inseparable" space and to show the existence of a family of closed sets such that each of them is of full measure, but their intersection is empty. The presented classic construction is credited to Jean Dieudonné and dates back to 1939. We also propose certain, up to our best knowledge, new simplifications.
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Czasopismo
Rocznik
Tom
Strony
76--84
Opis fizyczny
Bibliogr. 18 poz.
Twórcy
autor
- Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland
- Faculty of Cybernetics, Military University of Technology, ul. Gen. Witolda Urbanowicza 2, 00-908, Warsaw, Poland
Bibliografia
- [1] Dieudonné J., Un exemple d’un espace normal non susceptible d’une structure uniforme d’espace complet, C. R. Acad. Sci. Paris, 1939, 209, 145–147
- [2] Hessenberg G., Grundbegriffe der Mengenlehre: zweiter Bericht über das Unendliche in der Mathematik, Vandenhöck & Ruprecht, Göttingen, 1906
- [3] Ulam S., Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math., 1930, 16, 140–150
- [4] Alexandroff A. D., Additive set functions in abstract spaces, Mat. Sbornik: 1940, 8(50), 307–348; 1941, 9(51), 563–628; 1943, 13(55), 169–238
- [5] Rohlin V. A., On the fundamental ideas of measure theory, Mat. Sbornik, 1949, 25, 107–150
- [6] Marczewski E., On compact measures, Fund. Math., 1953, 40, 113–124
- [7] Marczewski E., Remarks on the convergence of measurable sets and measurable functions, Colloq. Math., 1955, 3, 118–124
- [8] Marczewski E., Collected mathematical papers, PAN, Warszawa, 1996
- [9] Bogachev V., Measure Theory, Volumes 1 and 2, Springer-Verlag Berlin Heidelberg, 2007
- [10] Kałamajska A., On Young measures controlling discontinuous functions, J. Conv. Anal., 2006, 13, 177–192
- [11] Kałamajska A., Kružík M., Oscillations and concentrations in sequences of gradients, ESAIM: COCV, 2008, 14, 71–104
- [12] Kružík M., Roubíček T., On the measures of DiPerna and Majda, Math. Bohem., 1997, 122, 383–399
- [13] Alibert J., Bouchitté G., Non-uniform integrability and generalized Young measures, J. Convex Anal., 1997, 4, 125–145
- [14] Chipot M., Kinderlehrer D., Equilibrium configurations of crystals, Arch. Rational. Mech. Anal., 1988, 103, 237–277
- [15] Kružík M., Prohl A., Young measure approximation in micromagnetics, Numer. Math., 2001, 90, 291–307
- [16] DiPerna R., Majda A., Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 1987, 108, 667–689
- [17] Moschovakis Y., Notes on Set Theory, Springer Science and Business Media, 2006
- [18] Hrbacek K., Jech T., Introduction to Set Theory, Third Edition, Revised and Expanded, CRC Press, 1999
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
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Bibliografia
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