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A counterexample to a Vitali type theorem with respect to Hausdorff content

Authors: Mark Melnikov and Joan Orobitg
Journal: Proc. Amer. Math. Soc. 118 (1993), 849-856
MSC: Primary 28A78
MathSciNet review: 1137228
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Abstract: Mateu and Orobitg proved (in Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990)) that given $ \lambda > 1$ and $ d - 1 < \alpha \leqslant d$ there exist constants $ C$ and $ N$ (depending on $ \lambda $ and $ \alpha $) with the following property:

For any compact set $ K$ in $ {\mathbb{R}^d}$ one can find a (finite) family of balls $ \{ B({x_i},{r_i})\} $ such that (i) $ K \subset \bigcup {B({x_i},{r_i})} $, (ii) $ \sum {r_i^\alpha \leqslant C{M^\alpha }(K)} $, $ {M^\alpha }$ denoting the $ \alpha $-dimensional Hausdorff content, and (iii) the dilated balls $ \{ B({x_i},\lambda {r_i})\} $ are an almost disjoint family with constant $ N$.

In this paper we prove that such a result is false for $ \alpha \leqslant d - 1$.

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Article copyright: © Copyright 1993 American Mathematical Society