Durrett's book Inversion Formula (Theorem 3.3.4)

Question

I'm reading Durrett's probability book. In page 109, after stating inversion formula it gives a counter example for point mass function. I'm a little bit confused. The theorem is not true for all probability measures?? So for which probability measures we can use the inversion formula?

Answer 1

The remark, taking $\mu = \delta_0$, says that the integrand is $$\frac{e^{-ita}-e^{-ibta}}{it} \varphi(t) = \frac{2\sin(t)}{t}$$ for $a=-1$ and $b=1$, i.e. this function is not integrable in the sense of Lebesgue. There are also other examples of this kind. Thus, in the inversion formula (theorem 3.3.4), that is $$\lim_{T \rightarrow \infty} \int_{-T}^T \frac{e^{-ita}-e^{-itb}}{it} \varphi(t) \, \mathrm{d} t = \mu((a,b)) + \frac{\mu \{a,b\}}{2},$$ we need to take the "symmetric limit", because the integrand is in general not Lebesgue-integrable.