Consider $f(x)$, a rapidly decreasing function, such that $\int_0^{\infty} f(x)=0$ and for $x$ near zero: $f(x)=O(x^a)$ (wit $a>0$). Can we interchange the sum and integral and write as below: $$\int_0^{\infty}\sum_{n=1}^{\infty} f(nx)= \sum_{n=1}^{\infty} \int_0^{\infty} f(nx) =0$$

Note that, as $\int_0^{\infty} f(x)=0$, the Poisson summation formula (thanks to the limit of $f(x)$ in zero) ensures that: $\sum_{n=1}^{\infty} f(nx)\sim O(x^a) \;\; (x \to 0)$ so the integral on the left in above expression is well defined.

We cannot apply directly the classical interchange theorems as: $\sum_{n=1}^{\infty} |f(nx)| \sim O(\frac{1}{x})$ and even if we have simple convergence of the sum, the partial sum $|\sum_{n=1}^{N} f(nx)|$ cannot be bounded near zero for all N (even if the complete sum is absolutely integrable). So is there way to show we can interchange or two sums are different, and if it is the case how can we show it ?