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Hausdorff and packing dimensions and sections of measures
Author(s) -
Järvenpää Maarit,
Mattila Pertti
Publication year - 1998
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300014042
Subject(s) - mathematics , hausdorff space , packing dimension , hausdorff distance , geometry , combinatorics , mathematical analysis , fractal , fractal dimension , minkowski–bouligand dimension
Let m and n be integers with 0< m < n and let μ be a Radon measure on ℝ n with compact support. For the Hausdorff dimension, dim H , of sections of measures we have the following equality: for almost all ( n − m )‐dimensional linear subspaces Vess inf {dim H μ V , a : a ∈ V ⊥ with μ V , a( ℝ n ) > 0 } = dim H μ − mprovided that dim H μ > m . Here μ v,a is the sliced measure and V ⊥ is the orthogonal complement of V . If the ( m + d )‐energy of the measure μ is finite for some d >0, then for almost all ( n − m )‐dimensional linear subspaces V we have ess inf {dim p μ V , a : a ∈ V ⊥ with μ V , a( ℝ n ) > 0 } = d μ .