Convergence of the real part of the integral associated with a characteristic function

Question

Let $I_n (t) = \int (e^{itu} -1)\frac{1+ u^2}{u^2}\, dG_n(u)$ be such that $$ I_n(t) \to \log f(t) $$

where $f(t)$ is the characteristic function of an infinitely divisible law.

Why is it that $$ \Re I_n(t) \to \log |f(t)|\;\;\;?$$

This is taken from Kolmogorov's book:

Answer 1

For complex numbers, by definition $\log z = \log |z| + i \arg z$. Consequently, since $I_n(t) \to \log f(t) = \log |f(t)| + i \arg f(t)$, where only the first term is real, $\Re I_n(t) \to \log |f(t)|$.