## 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.

### Citations:

Zbl 0858.28007; Zbl 0040.20804; Zbl 0791.28002
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