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.



Constructing measures with support in a given set
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.


