Packing dimension of mean porous measures

Details Packing dimension of mean porous measures Packing dimension of mean porous measures

2007-05-16

arxiv.org/abs/0705.2447v2

Mathematics

Classical Analysis and ODEs

Revised version

Scientific paper

We prove that the packing dimension of any mean porous Radon measure on $\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \cite{BS}, and in a weaker form in \cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\mu$ on $\mathbb R$ such that $\mu(A)=0$ for all mean porous sets $A\subset\mathbb R$.

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