Algebraically invariant extensions of -finite measures on Euclidean space

Author:
Krzysztof Ciesielski

Journal:
Trans. Amer. Math. Soc. **318** (1990), 261-273

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.

**[Be]**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****[CK]***Model theory*, Handbook of mathematical logic, Part A, North-Holland, Amsterdam, 1977, pp. 3–313. Studies in Logic and the Foundations of Math., Vol. 90. With contributions by Jon Barwise, H. Jerome Keisler, Paul C. Eklof, Angus Macintyre, Michael Morley, K. D. Stroyan, M. Makkai, A. Kock and G. E. Reyes. MR**0491125****[Ci]**Krzysztof Ciesielski,*How good is Lebesgue measure?*, Math. Intelligencer**11**(1989), no. 2, 54–58. MR**994965**, 10.1007/BF03023824**[CP]**Krzysztof Ciesielski and Andrzej Pelc,*Extensions of invariant measures on Euclidean spaces*, Fund. Math.**125**(1985), no. 1, 1–10. MR**813984****[GR]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[Ha1]**A. B. Harazisvili,*On Sierpinski's problem concerning strict extendibility of an invariant measure*, Soviet Math. Dokl.**18**(1977), 71-74.**[Ha2]**A. B. Kharazishvili,*Groups of transformations and absolutely negligible sets*, Soobshch. Akad. Nauk Gruzin. SSR**115**(1984), no. 3, 505–508 (Russian, with English and Georgian summaries). MR**797907****[Hu]**A. Hulanicki,*Invariant extensions of the Lebesgue measure*, Fund. Math.**51**(1962/1963), 111–115. MR**0142709****[Je]**Thomas Jech,*Set theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR**506523****[La]**Serge Lang,*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234****[MW]**Jan Mycielski and Stan Wagon,*Large free groups of isometries and their geometrical uses*, Enseign. Math. (2)**30**(1984), no. 3-4, 247–267. MR**767903****[Pe]**Andrzej Pelc,*Invariant measures and ideals on discrete groups*, Dissertationes Math. (Rozprawy Mat.)**255**(1986), 47. MR**872392****[Pk]**S. S. Pkhakadze,*teorii lebegovskoi miery*, Trudy Tbiliss. Mat. Inst.**25**(1958). (Russian)**[Ro]**Abraham Robinson,*Complete theories*, North-Holland Publishing Co., Amsterdam, 1956. MR**0075897****[Ru]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[Sz]**E. Szpilrajn,*Sur l'extension de la mesure lebesguienne*, Fund. Math.**25**(1935), 551-558. (French)**[We]**B. Węglorz,*Large invariant ideals on algebras*, Algebra Universalis**13**(1981), no. 1, 41–55. MR**631408**, 10.1007/BF02483821

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0946422-X

Keywords:
Invariant -finite measures,
algebraic transformations of ,
isometries of

Article copyright:
© Copyright 1990
American Mathematical Society