Singular continuous measures "in nature"

Question

According to the Lebesgue decomposition theorem, there are 3 basic kinds of measures on $\mathbb R^n$: continuous measures (those with a density), discrete measures, and singular continuous measures (those supported on a Lebesgue-null set, but with a continuous cdf).

The first two kinds are obviously encountered in countless contexts. However, the only examples of the third kind that I am familiar with are rather artificial (e.g., the Cantor measure).

Would a practicing statistician ever have occasion to work with a singular continuous measure?

Answer 1

Any transformation that makes you loose some dimension will transform a continuous measure into a type three measure. For example if you have a measure in $\mathbb R^n$, the measure inferred by the transformation $\mathbf x \rightarrow \frac{\mathbf x}{\lVert \mathbf x \rVert}$ will have a support that is Lesbegue-null in $\mathbb R^n$. The same will happen with any linear transformation from $\mathbb R^n$ to $\mathbb R^n$ that is not onto (the $n\times n$ transformation matrix is not full rank).

These scenario seems kind of cheating about the problem but I think that what is bothering you is that the Cantor does not have integer dimension which makes it less intuitive to us.

Answer 2

There are results of Jitomirskaya,Makarov and Simon, showing that for a suitably "generic" operator on a Hilbert space, the spectrum is singular continuous. This of course has relevance to quantum mechanics, in which the Hamiltonian of a system is a self-adjoint operator, and the spectrum corresponds to energy level. So in this sense, singular continuous measures are the default.