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1997/1998 A Typical Measure Typically Has No Local Dimension
Julia Genyuk
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Real Anal. Exchange 23(2): 525-538 (1997/1998).


We consider local dimensions of probability measure on a complete separable metric space \(X\): \(\overline{\alpha}_\mu(x) =\underset{r\to 0}{\varlimsup} \frac{\log_\mu(B_r(x))}{\log r}, \underline{\alpha}_\mu(x)=\underset{r\to 0}{\varliminf} \frac{\log_\mu(B_r(x))}{\log r}\). We show (Theorem 2.1) that for a typical probability measure \(\underline{\alpha}_\mu(x)=0\) and \(\overline{\alpha}_\mu(x)=\infty\) for all \(x\) except a set of first category. Also \(\underline{\alpha}_\mu(x)=0\) almost everywhere and with some additional conditions on \(X\) there is a corresponding result for upper local dimension: in particular, we show that a typical measure on \([0,1]^d\) has \(\overline{\alpha}_\mu(x)=d\) almost everywhere (Theorem 2.4). There are similar results concerning ``global'' dimensions of probability measures. Theorems 2.2 and 2.3 show in particular that the Hausdorff dimension of a typical measure on any compact separable space equals 0 and the packing dimension of a typical measure on \([0,1]^d\) equals \(d\).


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Julia Genyuk. "A Typical Measure Typically Has No Local Dimension." Real Anal. Exchange 23 (2) 525 - 538, 1997/1998.


Published: 1997/1998
First available in Project Euclid: 14 May 2012

zbMATH: 0943.28008
MathSciNet: MR1639964

Primary: 28A33 , 28A80

Keywords: {category} , {dimension} , {measure}

Rights: Copyright © 1999 Michigan State University Press

Vol.23 • No. 2 • 1997/1998
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