TITLE: What is an example of a measure without a density that is useful for some application?
QUESTION [9 upvotes]: I've been self-studying measure theory from Richard Ash's textbook, and am close to completing Chapter 2 (so, for instance, I've covered the Radon-Nikodym Theorem).
What is an example of a measure which is useful for some application for which there is not a density with respect to either counting measure or Lebesgue measure?
I would be especially interested to see an example that is a probability measure.
I've seen it claimed (e.g. by David Pollard) that such examples are prevalent in practice (and that the intuition of statisticians can be unnecessarily restricted by considering probability distributions merely in terms of those with probability density functions or probability mass functions).
But I can think of no such example at this point in my study.

REPLY [4 votes]: As you mention you studied the Radon-Nikodym theorem, it is a good exercise if you haven't already seen the proof to try and prove the following useful fact.
Let $\nu$ stand for some fixed reference Borel measure on $\mathbb R^n$. For any Borel measure $\mu$ on $\mathbb R^n$, there exist Borel measures $\mu_{sing},\mu_{ac}$ such that $\mu = \mu_{sing} + \mu_{ac}$. Moreover, $\mu_{sing}$ is singular with respect to $\nu$, and $\mu_{ac}$ is absolutely continuous with respect to $\nu$. In applications of Fourier analysis to partial differential equations, considering the Fourier transform $\widehat{\mu}$ of certain measures is a powerful idea, and for linear PDE, the decomposition into singular and absolutely continuous parts (with respect to some reference measure) allows us to study $\widehat\mu_{sing}$ and $\widehat\mu_{ac}$ somewhat separately.
Remark: As an example, if $f$ is an absolutely continuous function, then $df/dx$ exists almost everywhere and defines a signed measure on $\mathbb R$. It's interesting to consider whether there are any criteria that tell us when $df/dx$ has zero singular part, and when $df/dx$ defines a positive measure.