Algebraically invariant extensions of finite measures on Euclidean space
Author:
Krzysztof Ciesielski
Journal:
Trans. Amer. Math. Soc. 318 (1990), 261273
MSC:
Primary 28C10
MathSciNet review:
946422
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Abstract: Let be a group of algebraic transformations of , i,e., the group of functions generated by bijections of of the form where each is a rational function with coefficients in in variables. For a function we say that a measure on is invariant when for every and every measurable set . We will examine the question: "Does there exist a proper invariant extension of We prove that if is finite then such an extension exists whenever contains an uncountable subset of rational functions such that for all . In particular if is any uncountable subgroup of affine transformations of is the absolute value of the Jacobian of and is a invariant extension of the dimensional Lebesgue measure then has a proper invariant extension. The conclusion remains true for any finite measure if is a transitive group of isometries of . An easy strengthening of this last corollary gives also an answer to a problem of Harazisvili.
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 C. Chang and H. Keisler, Model theory, Studies in Logic and Foundations of Math., NorthHolland, 1977. MR 0491125 (58:10395)
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 R. C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, Englewood Cliffs, N.J., 1965. MR 0180696 (31:4927)
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 A. B. Harazisvili, On Sierpinski's problem concerning strict extendibility of an invariant measure, Soviet Math. Dokl. 18 (1977), 7174.
 [Ha2]
 , Groups of transformations and absolutely negligible sets, Bull. Acad. Sci. Georgian SSR 115 (1984). (Russian) MR 797907 (86h:54046)
 [Hu]
 A. Hulanicki, Invariant extensions of the Lebesgue measure, Fund. Math. 51 (1962), 111115. MR 0142709 (26:278)
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 T. Jech, Set theory, Academic Press, 1978. MR 506523 (80a:03062)
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 S. Lang, Algebra, AddisonWesley, 1984. MR 0197234 (33:5416)
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 J. Mycielski and S. Wagon, Large free groups of isometries and their geometrical uses, Enseign. Math. 30 (1984), 247267. MR 767903 (86a:20042)
 [Pe]
 A. Pelc, Invariant measures and ideals on discrete groups, Dissertationes Math. 255 (1986). MR 872392 (88b:28025)
 [Pk]
 S. S. Pkhakadze, teorii lebegovskoi miery, Trudy Tbiliss. Mat. Inst. 25 (1958). (Russian)
 [Ro]
 A. Robinson, Complete theories, NorthHolland, Amsterdam, 1956. MR 0075897 (17:817b)
 [Ru]
 W. Rudin, Real and complex analysis, McGrawHill, 1987. MR 924157 (88k:00002)
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 E. Szpilrajn, Sur l'extension de la mesure lebesguienne, Fund. Math. 25 (1935), 551558. (French)
 [We]
 B. Weglorz, Large invariant ideals on algebras, Algebra Universalis 13 (1981), 4155. MR 631408 (83f:04004)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719900946422X
PII:
S 00029947(1990)0946422X
Keywords:
Invariant finite measures,
algebraic transformations of ,
isometries of
Article copyright:
© Copyright 1990
American Mathematical Society
