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Fractal properties of products and projections of measures in ℝd

Published online by Cambridge University Press:  24 October 2008

Xiaoyu Hu
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
S. James Taylor
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.

Abstract

Borel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1994

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