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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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