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

Kharazishvili, A. B.

Acta Univ. Carol., Math. Phys. 35, No. 2, 33-39 (1994).

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)]}.

Reviewer: Z.Lipecki (Wrocław)

MSC:

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

Keywords:

invariant measure; extension; uniqueness property; Steinhaus property; Lebesgue measure; density point

Citations:

Zbl 0858.28007; Zbl 0040.20804; Zbl 0791.28002