平均多孔测度的填充维数
Packing dimension of mean porous measures
摘要：
证明了$\mathbb R^d$上任意平均孔隙度Radon测度的堆积维数都可以由一个依赖于平均孔隙度的函数来估计。当平均孔隙度趋于其最大值时，上界趋于$d-1$。这个结果在\cite{BS}中陈述过，在\cite{JJ1}中以较弱的形式陈述过，但是证明是不正确的。相当令人惊讶的是，原来平均多孔措施不一定是近似的平均多孔集。我们通过构建一个平均多孔措施$μ$ $\mathbb R$的例子，使$μ(A)=0$的所有平均多孔集$A\subset\mathbb R$验证这一点。
英文摘要：
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 $μ$ on $\mathbb R$ such that $μ(A)=0$ for all mean porous sets $A\subset\mathbb R$.
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