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.

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DOI:
http://dx.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