I am willing to get an idea of what is, for $\alpha >1$, $$\int_0^\infty \sum_{n \in (x,\alpha x)} f(n)\frac{dx}{x}$$
for any (positive) function $f$. I would like to say that essentially we get the full sum over $n$, but of course there should be some overlapping (e.g. if I double $\alpha$ I would expect that the value of the integral... double?)
However I do not succeed in formalizing this properly, is there any standard statement of this result? Should I prove it explicitly with characteristic functions?
$$\int_0^{\infty} \sum_n \chi_{x<n<\alpha x} \frac {f(n)} x dx$$ $$=\sum_n \int_0^{\infty} \chi_{\frac n {\alpha} <x<n} \frac {f(n)} x dx$$ $$=\sum_n [\ln (n)-\ln ( \frac n {\alpha} )] f(n)$$ $$ =\ln (\alpha) \sum f(n)$$.
The first equality is a consequence of Fubini/Tonelli Theorem or Monotone Convergence Theorem.