A note on the generalized fractal dimensions of a probability measure

We prove the following result on the generalized fractal dimensions $D^{±}_q$ of a probability measure $\mu$ on $R^n$. Let $g$ be a complex-valued measurable function on $R^n$ satisfying the following conditions: (1) $g$ is rapidly decreasing at infinity, (2) $g$ is continuous and nonvanishing at (at least) one point, (3) $\int g≠0$. Define the partition function $\Lambda_a(μ,q)=a^{n(q−1)}‖g_a * μ‖\lim_q q$, where $g_a(x)=a^{−n}g(a^{−1}x)$ and $*$ is the convolution in $R^n$. Then for all $q>1$ we have $D^{±}_q=1/(q−1)\lim_{r→0} {}^{sup}_{inf}[\log \Lambda_a \mu(r,q) / \log r]$