TITLE: Singular distributions: Applications and Instances
QUESTION [6 upvotes]: Singular distributions are special mathematical objects. They have an interesting property of not having a density function, defined on a set with Lebesgue measure zero. Cantor distribution is the typical example of such distribution. 
I was wondering whether there are instances in nature that are explained using singular distributions or any applications of singular distribution in engineering milieu.
One interesting observation about Cantor set is that it is a self-similar set, like Sierpinski triangle and Koch snowflake. Therefore one candidate is obviously fractals. 
My personal Google inquiry resulted in some papers and works in the fractal electro-dynamics theory and even economics. But I prefer to hear from experts on each domain.
I also wonder how a phenomenon related to singular distributions differs or is expected to differ from those with non-singular ones.

REPLY [3 votes]: My colleagues and collaborators, Lutwak and Zhang, have worked together with others to solve Minkowski type problems, where a Borel measure is given on the unit sphere and one solves for a convex body for which a certain geometric measure is equal to the given Borel measure. For continuous measures, these problems always have a solution, but when the measure is singular, there are obstructions.