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Hausdorff dimension, Borel maps and random sets

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If you have a question about this talk, please contact Neal Bez.

Is there a Borel map f: R → R that takes all “small” sets to “large” sets? Can one increase the dimension of all sets by a Borel map? Are the 1/3 and 1/2-dimensional Hausdorff measures essentially the same, that is, isomorphic?

To answer the above questions we will consider some constructions of random compact sets and give dimension estimates. We will show that Hausdorff measures of different dimensions are not Borel isomorphic, solving a problem of D.Preiss and B.Weiss.

This talk is part of the Analysis seminar series.

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