What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = \left(\sum_{k=1}^\infty k^{-1-p}\right)\left( \int_\mathbb{R}|f(y)|dy\right) < \infty. $$ The first equality is by a corollary to Monotone Convergence theorem, the second by change of variable, and the inequality by integrability of $f$and convergence of the numerical series.
So $\sum_{k=1}^\infty k^{-p}|f(kx)| < \infty$ almost everywhere. The problem is that I haven't gotten to change-of-variable theorem yet and should not use it.
If you could please show a way to do this without using change of variable, thank you!