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I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm primarily interested in $n=1$ and compact subsets).

I'm interested in finding similar results, i.e. starting with some subset of $\mathbb{R}$ ($\mathbb{R}^n$), maybe some additional parameters and then constructing a measure with specific properties on that set.

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I think the question is too broad. What are "specific properties"? All sorts of equilibrium measures from potential theory quality.

Here is an example of a deep result: on every compact set in $R^n$ there exists a non-zero measure satisfying the doubling condition, which means that the measure of every ball is at most constant times the measure of twice smaller ball with the same center. MR0765294.

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  • $\begingroup$ Thanks, that example is precisely the kind of result I'm looking for. As I am not planning to invoke such a result, I can't really specify more what I want. $\endgroup$
    – Arno
    Commented Feb 18, 2014 at 1:15

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