Local dimensions of sliced measures and stability of packing dimensions of sections of sets

Let m and it be integers with 0<m<n. We relate the absolutely continuous and singular parts of a measure mu on R-n to certain properties of plane sections of mu. This leads us to prove, among other things, that the lower local dimension of (n - m)-plane sections of mu is typically constant provided that the Hausdorff dimension of mu is greater than in. The analogous result holds for the upper local dimension if mu has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets. (C) 2003 Elsevier Science (USA). All rights reserved.