ON THE CENTRED HAUSDORFF MEASURE

Abstract

Let v be a measure on a separable metric space. For t, q ∈ R, the centred Hausdorff measures μ^h with the gauge function h(x, r) = r^t(vB(x, r))^q is studied. The dimension defined by these measures plays an important role in the study of multifractals. It is shown that if v is a doubling measure, then μ^h is equivalent to the usual spherical measure, and thus they define the same dimension. Moreover, it is shown that this is true even without the doubling condition, if q [ges ] 1 and t [ges ] 0 or if q [les ] 0. An example in R² is also given to show the surprising fact that the above assertion is not necessarily true if 0 < q < 1. Another interesting question, which has been asked several times about the centred Hausdorff measure, is whether it is Borel regular. A positive answer is given, using the above equivalence for all gauge functions mentioned above.