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# Mathematics Colloquium: Doubling Measures

2010-09-14

4:10 pm; Neill Hall 5W

John Garnett

Abstract: A positive regular Borel measure Î½ on a metric space (X, d) is a doubling measure if there is a constant C such that for all x in X and all r > 0, Î½(B(x,2r)) â‰¤ CÎ½(B(x,r)), where B(x, r) = {y : d(y, x) < r}. Doubling measures have many applications in analysis. I will survey some of these applications and then consider the problem: Problem: When does E (a subset of X) have Î½ measure zero for all doubling measures Î½ on X? For example every line has measure zero for all doubling measures on R2. I will explain an example, due jointly with Rowan Killip and Raanan Schul, of a rectifiable curve in Î“ (a subset of R2) and a doubling measure Î½ on R2 such that Î½ (Î“) > 0.