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

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.


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