Let \((X,\rho)\) be a complete separable metric space and \(\mathcal M\) be the set of all probability Borel measures on \(X\). We show that if the space \(\mathcal M\) is equipped with the weak topology, the set of measures having the upper (resp. lower) correlation dimension zero is re\-si\-dual. Moreover, the upper correlation dimension of a typical (in the sense of Baire category) measure is estimated by means of the local lower entropy and local upper entropy dimensions of \(X\).
"Typical properties of correlation dimension.." Real Anal. Exchange 28 (2) 269 - 278, 2002/2003.