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.