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 \(d\).


28A78 Hausdorff and packing measures
28A80 Fractals