I want to show $\sum_{n=0}^\infty \int |f_n(x)|dx$ converges if $f$ is integrable with $f_n(x)=f(a_nx)$ and $\sum_{n=0}^\infty \frac{1}{a_n}$ converges. Here, $a_n$ is positive and integral is on $\mathbb{R}$.
I try to use the theorems in my course, so I think I can permute the sum and the integral, but I don't have many informations on $f_n$ so it's a little bit difficult. Can you help me ?