HAUSDORFF AND PACKING DIMENSIONS AND SECTIONS OF MEASURES

Let m and n be integers with 0 < m <n and let mu be a Radon measure on R-n with compact support. For the Hausdorff dimension, dim(H), of sec tions of measures we have the following equality: for almost all (n - m)- dimensional linear subspaces V ess inf {dim(H) mu (V,a) : a is an element of V-perpendicular to with mu(V,a)(R-n) > 0} = dim(H) mu - m p rovided that dim(H) > m. Here mu(V,a) is the sliced measure and V-perp endicular to is the orthogonal complement of V. If the (m+d)-energy of the measure mu is:finite for some d > 0, then for almost all (n - m)- dimensional linear subspaces V we have ess inf {dim(p) mu(V,a) : a is an element of V-perpendicular to with mu(V,a) (R-n) > 0} = d(mu). Here dim, is the packing dimension and d, is a constant defined by means o f the convolution of mu with a certain kernel. We also deduce correspo nding results for the upper packing and upper Hausdorff dimensions.