Premium
On the Centred Hausdorff Measure
Author(s) -
Schechter Alexander
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001356
Subject(s) - hausdorff measure , dimension function , mathematics , measure (data warehouse) , borel measure , hausdorff dimension , separable space , equivalence (formal languages) , outer measure , function (biology) , pure mathematics , dimension (graph theory) , combinatorics , metric space , hausdorff space , minkowski–bouligand dimension , discrete mathematics , mathematical analysis , probability measure , fractal , fractal dimension , database , evolutionary biology , biology , computer science
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 ⩾ 1 and t ⩾ 0 or if q ⩽ 0. An example in R 2 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.