On equivalent definitions of the correlation dimension for a probability measure

Abstract

In mathematical physics, one sometimes has to deal with averages of the type

$$M\mu (T) = \frac{1}{{T^n }}\int\limits_{|\xi | \leqslant T} { d\xi |\hat \mu (\xi )|^2 , T > 0}$$

where$\hat \mu$ is the Fourier transform of some probability Borel measure μ. We show that the asymptotic behavior ofMμ is governed by the usual (upper and lower) correlation dimension of the measure μ.