Access
You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen Reader
This content is available through Read Online (Free) program, which relies on page scans. 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.Journal Article
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. 419430
Published by: American Mathematical Society
DOI: 10.2307/2046989
Stable URL: http://www.jstor.org/stable/2046989
Page Count: 12
You can always find the topics here!
Topics: Lebesgue measures, Rectangles, Mathematical intervals, Mathematical theorems, Factorization, Integers, Average linear density, Continuous functions, Increasing functions, Algebra
Were these topics helpful?
See something inaccurate? Let us know!
Select the topics that are inaccurate.
 Item Type
 Article
 Thumbnails
 References
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.
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 Cantortype 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 $I$.
Page Thumbnails

419

420

421

422

423

424

425

426

427

428

429

430
Proceedings of the American Mathematical Society © 1988 American Mathematical Society