A TYPICAL MEASURE TYPICALLY HAS NO LOCAL DIMENSION
A TYPICAL MEASURE TYPICALLY HAS NO LOCAL DIMENSION
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 …