## Real Analysis Exchange

### A Typical Measure Typically Has No Local Dimension

Julia Genyuk

#### Abstract

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 $d$.

#### Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 525-538.

Dates
First available in Project Euclid: 14 May 2012

https://projecteuclid.org/euclid.rae/1337001362

Mathematical Reviews number (MathSciNet)
MR1639964

Zentralblatt MATH identifier
0943.28008

Keywords
{measure} {dimension} {category}

#### Citation

Genyuk, Julia. A Typical Measure Typically Has No Local Dimension. Real Anal. Exchange 23 (1999), no. 2, 525--538. https://projecteuclid.org/euclid.rae/1337001362