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Some remarks on density points and the uniqueness property for invariant extensions of the Lebesgue measure. (English) Zbl 0858.28008

The author constructs a \(\sigma\)-algebra \(S\) of subsets of \(\mathbb{R}\) and a measure \(\mu\) on \(S\) such that: 1. \(S\) contains the Lebesgue \(\sigma\)-algebra of \(\mathbb{R}\) and \(\mu_1\) extends Lebesgue measure; 2. Both \(S\) and \(\mu_1\) are invariant under the isometries of \(\mathbb{R}\); 3. \(\mu_1\) is a unique up to multiplicative constant \(\sigma\)-finite measure on \(S\) which is invariant under the isometries of \(\mathbb{R}\); 4. There exists \(A\in S\) with a unique density point with respect to the family of open intervals on \(\mathbb{R}\). The construction of \(S\) and \(\mu_1\) uses a discontinuous character of \(\mathbb{R}\) and is taken essentially from a paper by K. Kodaira and S. Kakutani [Ann. Math., II. Ser. 52, 574-579 (1950; Zbl 0040.20804)]. Property 3 follows from a general uniqueness result the proof of which is given in a book by the author [“Vitali’s theorem and its generalizations” (1991; Zbl 0791.28002), Theorem 19 of Section 5]. The definition of \(A\) uses a geometric idea. Moreover, the author announces that \(\mu_1\) lacks what he calls the Steinhaus property. This means that for some \(B\in S\) and a sequence \((x_n)\) in \(\mathbb{R}\) with \(x_n\to 0\) we have \(\mu_1((B+ x_n)\cap B)\nrightarrow\mu_1(B)\).
{Reviewer’s remark: The announced result is proved in another paper by the author (see the preceding review). Related material can be found in a subsequent paper by the author [Real Anal. Exch. 21, 743-749 (1996)]}.

MSC:

28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
43A05 Measures on groups and semigroups, etc.
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