Some properties of Buck’s measure density. (English) Zbl 0761.11003

Paštéka, Milan

Math. Slovaca 42, No. 1, 15-32 (1992).

For any set \(A\subseteq\mathbb{N}\) let \(\Gamma(A)=\sum_{a\in A}2^{-a}\); let \(D_0:=\{A\subseteq\mathbb{N}\mid\) \(\Gamma(A)\in\mathbb{Q}\}\) (“rational sets”). R. C. Buck [Am. J. Math. 68, 560–580 (1946; Zbl 0061.07503)] investigated asymptotic density as a content on \(D_ 0\), extending it to an outer measure \(\mu^*\) on \({\mathfrak P}(\mathbb{N})\) and introducing the class \(D_ 1\) of “\(\mu^*\)-measurable” sets (called “pseudorational” by the reviewer [J. Reine Angew. Math. 190, 199–230 (1952; Zbl 0048.03401)] and subsequently, by others).
The author proves some arithmetic as well as metric results on the class \(D_1\) and on Buck’s finitely, but not \(\sigma\)-additive density measure \(\mu=\mu^*\mid_{D_1}\), including the following assertions:
(i) \(\forall A\in D_1,\;\forall\alpha\in[0,\mu(A)]\quad \exists B\in D_1\), \(B\subseteq A\), \(\mu(B)=\alpha\).
(ii) \(\exists A\): \(\forall\alpha\in[0,1] \quad\exists S\subseteq A\): \(\mu^*(S)=\alpha\).
(iii) \(\mu^*(A)=1\Leftrightarrow A\) can be rearranged into a sequence which is u.d. in \(\mathbb{Z}\).
The author appears to be unaware of the fact that Hausdorff \(\dim\Gamma(D_1)=0\), proved by the reviewer [J. Reine Angew. Math. 193, 126–128 (1954; Zbl 0056.05103)], since he states a much weaker result on p. 27.

Reviewer: Bodo Volkmann (Stuttgart)

MSC:

11B05	Density, gaps, topology
28E99	Miscellaneous topics in measure theory
11B83	Special sequences and polynomials

Keywords:

sets of integers; Buck measure density; pseudorational sets; \(\mu^*\)-measurable sets; rational sets; asymptotic density; outer measure

Citations:

Zbl 0761.11004; Zbl 0061.07503; Zbl 0048.03401; Zbl 0056.05103

References:

[1]	BUCK R. C.: The measure theoretic approach to the density. Amer. J. Math. LXVIII (1946), 560-580. · Zbl 0061.07503
[2]	DIJKSMA A., MEIJER H. G.: Note on uniformly distributed sequences. Nieuw Arch. Wisk. (4) XVII (1969), 210-213. · Zbl 0186.08701
[3]	HLAWKA E.: Theorie der Gleichverteilung. Bibliographishes Institut, Manheim, Wien, Ztrich, 1979. · Zbl 0406.10001
[4]	KUIPERS Z., NIEDERREITER H.: Uniform distribution of sequences. John Wiley et Sons, New York-London Sydney Toronto, 1974. · Zbl 0281.10001
[5]	OXTOBY J. C.: Measure and Category. (Russian translation Moskva, Mir 1971), Springer, Berlin, 1971. · Zbl 0217.09201
[6]	ŠALÁT T.: Cantorsche Entwicklungen der reallen Zahlen und das Hausdorffsche Mass. Publ. of the Math. Inst, of the Hung. Acad, of Sci VI (1961), 15-41. · Zbl 0119.28502
[7]	PAŠTÉKA M., ŠALÁT T.: Buck’s measure density and sets of positive integers containing arithmetic progressions. Math. Slovaca 41 (1991), 283-293. · Zbl 0761.11004