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One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts:

$\mu_{ac}$: absolutely continuous

$\mu_{sc}$: singular continuous

$\mu_{pp}$: pure point

A common example for a singular continuous probability measure is Cantor's function as cdf. Such a cdf is continuous. I have two question:

(1) Do singular cont. probability measures come up? E.g. as law of pure jump L\'evy processes, for which there are criteria to guarantee a density. What about semimartingales?

(2) The characteristic function can be defined in two ways $\int e^{itx}dF(x)$ or $\int e^{itx}d\mu_{sc}(x)$. Though there is no density function, can $\int e^{itx}dF(x)$ be approximated by a sequence of, say a continuous $\mathcal{L}^1$ functions, such that $\int e^{itx}dF(x) = \lim_{n\rightarrow\infty}\int e^{itx}f_n(x)dx$?

I can't find anything on that, so maybe singular measures are quite opaque objects in probability.

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For problem $(1)$, I will quote from the book. An Introduction to Probability Theory and Its Applications by Feller.

Every atomic distribution is singular with respect to $dx$, but the Cantor distribution of example $1,11 (d)$ shows that there exist continuous distributions in $\mathbb{R}$ that are singular with respect to $dx$. Such distributions are not tractable by the methods of calculus and explicit representations are in practice impossible. For analytic purposes one is therefore forced to choose a framework which leads to absolutely continuous or atomic distributions. Conceptually, however, singular distributions play an important role and many statistical tests depend on their existence. This situation is obscured by the cliche that "in practice" singular distributions do not occur.