THE AVERAGE FRACTAL DIMENSION AND PROJECTIONS OF MEASURES AND SETS INR(N)

In this note we introduce the concept of local average dimension of a measure mu at x is an element of R(n) as the unique exponent where the lower average density of mu at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of mu, respectively. The average dimension of an a nalytic set E is defined as the supremum over the upper average dimens ions of all measures supported by E. These average dimensions lie betw een the corresponding Hausdorff and packing dimensions and the inequal ities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of mu at almost all x are invariant under orthogonal projections onto almost all m-dimensional linear subspaces of higher dimension. The corresponding global results for mu and E (w hich are known for Hausdorff dimension) follow immediately.