ON THE CENTRED HAUSDORFF MEASURE

Let v be a measure on a separable metric space. For t, q ∈ R, the centred Hausdorff measures μh with the gauge function h(x, r) = rt(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 μh 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.