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Factorizations of Lebesgue Measure via Convolutions

Norman Levenberg, Jr., Gaven J. Martin, Allen L. Shields and Smilka Zdravkovska
Proceedings of the American Mathematical Society
Vol. 104, No. 2 (Oct., 1988), pp. 419-430
DOI: 10.2307/2046989
Stable URL: http://www.jstor.org/stable/2046989
Page Count: 12
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Factorizations of Lebesgue Measure via Convolutions
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Abstract

Given a continuous, increasing function $\phi: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ with $\phi(0) = 0$, we define the Hausdorff $\phi$-measure of a bounded set $E$ in the unit interval $I = \lbrack 0, 1 \rbrack$ as $H_\phi(E) = \lim_{\delta \rightarrow 0} H_{\phi, \delta}(E)$ where $H_{\phi, \delta} E = \inf \sum^\infty_{i = 1} \phi(t_i)$ and the infimum is taken over all countable covers of $E$ by intervals $U_i$ with $t_i = |U_i| =\quad\text{length of}\quad U_i < \delta$. We show that given any such $\phi$, there exist closed, nowhere dense sets $E_1, E_2 \subset I$ with $H_\phi(E_1) = H_\phi(E_2) = 0$ and $E_1 + E_2 \equiv \{a + b: a \in E_1, b \in E_2 \} = I$. The sets $E_i (i = 1, 2)$ are constructed as Cantor-type sets $E_i = \bigcap^\infty_{n = 1} E_{i, n}$ where $E_{i, n}$ is a finite union of disjoint closed intervals. In addition, we give a simple geometric proof that the natural probability measures $\mu_i$ supported on $E_i$ which arise as weak limits of normalized Lebesgue measure on $E_{i, n}$ have the property that the convolution $\mu_1 \ast \mu_2$ is Lebesgue measure on $I$.

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