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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Factorizations of Lebesgue measure via convolutions


Authors: Norman Levenberg, Gaven J. Martin, Allen L. Shields and Smilka Zdravkovska
Journal: Proc. Amer. Math. Soc. 104 (1988), 419-430
MSC: Primary 28A50; Secondary 28A35
MathSciNet review: 962808
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Abstract: Given a continuous, increasing function $ \phi :[0,\infty ) \to [0,\infty )$ with $ \phi (0) = 0$, we define the Hausdorff $ \phi $-measure of a bounded set $ E$ in the unit interval $ I = [0,1]$ as $ {H_\phi }(E) = {\lim _{\delta \to 0}}{H_\phi }{,_\delta }(E)$ where $ {H_\phi }{,_\delta }E = \inf \sum\nolimits_{i = 1}^\infty {\phi ({t_i})} $ and the infimum is taken over all countable covers of $ E$ by intervals $ {U_i}$ with $ {t_i} = \left\vert {{U_i}} \right\vert = $ length of $ {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 \left\{ {a + b:a \in {E_1},b \in {E_2}} \right\} = I$. The sets $ {E_i}(i = 1,2)$ are constructed as Cantor-type sets $ {E_i} = \bigcap\nolimits_{n = 1}^\infty {{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}*{\mu _2}$ is Lebesgue measure on $ I$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0962808-8
Article copyright: © Copyright 1988 American Mathematical Society