## A typical measure typically has no local dimension.(English)Zbl 0943.28008

Summary: We consider local dimensions of probability measure on a complete separable metric space $$X:\overline {\alpha }_{\mu } (x) = \overline {\lim }_{r\rightarrow 0} \frac {\log \mu (B_r(x))}{\log r}$$, $$\underline {\alpha }_{\mu }(x) = \underline {\lim }_{r\rightarrow 0} \frac {\log \mu (B_r(x))}{\log r}$$. 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 $$d$$.

### MSC:

 28A78 Hausdorff and packing measures 28A80 Fractals