Background: reading part 2 to this answer, I want to comment asking what the name is for the distribution where $\mu(a,b)+\frac 12\mu(\{a,b\}) = \lim_{n\to +\infty}(2\pi)^{-1}\sum_{j=-n}^{n}\int_0^1\frac{e^{-i2j\pi b}\sin(tb)-e^{-i2j\pi a}\sin(ta)}{it}(1-t)\mathrm{d}t$.
Instead, I'd like to know if there's a simple example of a characteristic function that is non-integrable, but using the inversion formula:
$$\mu(a,b)+\frac 12\mu(\{a,b\})=\lim_{T\to +\infty}(2\pi)^{-1}\int_{-T}^T\frac{e^{-ita}-e^{-itb}}{it}\varphi(t)dt$$
results in an expression that looks similar to a CDF taught in an introductory probability course?
(Otherwise, I feel like the above form is not very practical compared to the form that assumes integrability of the characteristic function and computes the density.)