Variations on Lusin’s theorem

Brown, Jack B.; Prikry, Karel

doi:10.1090/S0002-9947-1987-0887497-6

Variations on Lusin’s theorem
HTML articles powered by AMS MathViewer

by Jack B. Brown and Karel Prikry PDF

Trans. Amer. Math. Soc. 302 (1987), 77-86 Request permission

Abstract:

We prove a theorem about continuous restrictions of Marczewski measurable functions to large sets. This theorem is closely related to the theorem of Lusin about continuous restrictions of Lebesgue measurable functions to sets of positive measure and the theorem of Nikodym and Kuratowski about continuous restrictions of functions with the Baire property (in the wide sense) to residual sets. This theorem is used to establish Lusin-type theorems for universally measurable functions and functions which have the Baire property in the restricted sense. The theorems are shown (under assumption of the Continuum Hypothesis) to be "best possible" within a certain context.

References

H. D. Block and Buchanan Cargal, Arbitrary mappings, Proc. Amer. Math. Soc. 3 (1952), 937–941. MR 52095, DOI 10.1090/S0002-9939-1952-0052095-5
Henry Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), no. 2, 113–128. MR 1501216, DOI 10.1090/S0002-9947-1922-1501216-9

Un théorème sur les ensembles measurables

137

J. C. Bradford and Casper Goffman, Metric spaces in which Blumberg’s theorem holds, Proc. Amer. Math. Soc. 11 (1960), 667–670. MR 146310, DOI 10.1090/S0002-9939-1960-0146310-1
Jack B. Brown, Metric spaces in which a strengthened form of Blumberg’s theorem holds, Fund. Math. 71 (1971), no. 3, 243–253. MR 292024, DOI 10.4064/fm-71-3-243-253
Jack B. Brown, Lusin density and Ceder’s differentiable restrictions of arbitrary real functions, Fund. Math. 84 (1974), no. 1, 35–45. MR 335699, DOI 10.4064/fm-84-1-35-45
Jack B. Brown, A measure theoretic variant of Blumberg’s theorem, Proc. Amer. Math. Soc. 66 (1977), no. 2, 266–268. MR 463377, DOI 10.1090/S0002-9939-1977-0463377-4

Variations on Blumberg’s theorem

9

Jack B. Brown, Continuous restrictions of Marczewski measurable functions, Real Anal. Exchange 11 (1985/86), no. 1, 64–71. The ninth summer real analysis symposium (Louisville, Ky., 1985). MR 828480
J. B. Brown and G. V. Cox, Classical theory of totally imperfect spaces, Real Anal. Exchange 7 (1981/82), no. 2, 185–232. MR 657320
Jack Ceder, Some examples on continuous restrictions, Real Anal. Exchange 7 (1981/82), no. 1, 155–162. MR 646647
Adam Emeryk, Ryszard Frankiewicz, and Włdysław Kulpa, On functions having the Baire property, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 6, 489–491 (English, with Russian summary). MR 560185
Ryszard Frankiewicz, On functions having the Baire property. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), no. 11-12, 559–560 (1983) (English, with Russian summary). MR 718734
Edward Grzegorek, Solution of a problem of Banach on $\sigma$-fields without continuous measures, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 1-2, 7–10 (1981) (English, with Russian summary). MR 616191
Edward Grzegorek, On some results of Darst and Sierpiński concerning universal null and universally measurable sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 1-2, 1–5 (English, with Russian summary). MR 634120

On sets always of the first category

Symmetric

fields of sets and universal null sets

E. Grzegorek, Always of the first category sets, Proceedings of the 12th winter school on abstract analysis (Srní, 1984), 1984, pp. 139–147. MR 782712
Edward Grzegorek and Czesław Ryll-Nardzewski, A remark on absolutely measurable sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 5-6, 229–232 (1981) (English, with Russian summary). MR 620193
E. Grzegorek and C. Ryll-Nardzewski, On universal null sets, Proc. Amer. Math. Soc. 81 (1981), no. 4, 613–617. MR 601741, DOI 10.1090/S0002-9939-1981-0601741-1

La propriété de Baire dans les espaces métriques

16

Kazimierz Kuratowski and Andrzej Mostowski, Set theory, Second, completely revised edition, Studies in Logic and the Foundations of Mathematics, Vol. 86, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN—Polish Scientific Publishers, Warsaw, 1976. With an introduction to descriptive set theory; Translated from the 1966 Polish original. MR 0485384

Sur une propriété des fonctions

137

Sur les propriétés des fonctions mesurables

Sur un classe de fonctions de M. Sierpinski et la classe correspondante d’ensembles

24

Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233. MR 776624
John C. Morgan II, Measurability and the abstract Baire property, Rend. Circ. Mat. Palermo (2) 34 (1985), no. 2, 234–244. MR 814038, DOI 10.1007/BF02850698

Sur la condition de Baire

Sur un problème de M. Ruziewicz concernant les superpositions de fonctions jouissant de la propriété de Baire

24

Sur une fonction qui est discontinue sur tout ensemble de puissance du continu

4

S. M. Srivastava, Selection theorems for $G_{\delta }$-valued multifunctions, Trans. Amer. Math. Soc. 254 (1979), 283–293. MR 539919, DOI 10.1090/S0002-9947-1979-0539919-3

Una proprietà delle funzioni misurabili

38

H. E. White Jr., Topological spaces in which Blumberg’s theorem holds, Proc. Amer. Math. Soc. 44 (1974), 454–462. MR 341379, DOI 10.1090/S0002-9939-1974-0341379-3
H. E. White Jr., Topological spaces in which Blumberg’s theorem holds. II, Illinois J. Math. 23 (1979), no. 3, 464–468. MR 537801