Packing dimension of mean porous measures

TL;DR

This paper establishes an upper bound on the packing dimension of mean porous measures in Euclidean space, showing it depends on mean porosity and approaches the space dimension minus one as porosity increases.

Contribution

It provides a rigorous proof for the packing dimension bound of mean porous measures, correcting previous incorrect proofs and demonstrating that such measures are not necessarily approximable by mean porous sets.

Findings

Upper bound on packing dimension depends on mean porosity

Bound approaches d-1 as mean porosity reaches maximum

Existence of mean porous measure not approximable by mean porous sets

Abstract

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 $A\subset\mathbb R$.