I've come across a statement in my book, that I think the author expected to be obvious so I feel like I'm missing something rather trivial here.
Suppose $f \in L_\text{loc}^1(\mathbb{R}^n)$. If $$\int_{|x| \leq A} |f(x)| \ dx \leq cA^N $$ , for some constants $c$ and $N$, then for any $\psi \in S(\mathbb R^n)$, we have that $$\int_{\mathbb{R}^n} |f(x)\psi(x)| \ dx < \infty$$ where $S(\mathbb{R}^n)$ is of course the Schwartz Class. Moreover, we have that this condition is necessary if $f$ is positive.
I don't see why this is true.
For $l\geqslant 1$, let $E_l:=\left\{x\in\mathbb R^n,l\lt \lVert x\rVert \leqslant l+1\right\}$ and let $M:=\sup_{x\in\mathbb R^n}\lVert x\rVert^{N+2}\left\lvert\phi(x)\right\rvert $. Then $$\int_{E_l}|f(x)\psi(x)| \mathrm dx=\int_{E_l}\lvert f(x)\rvert\underbrace{ \lvert \psi(x)\rvert \lVert x\rVert^{N+2}}_{\leqslant M} \cdot \underbrace{\lVert x\rVert^{-N-2}}_{\leqslant l^{-N-2}} \mathrm dx\leqslant Mc\frac{(l+1)^N}{l^{N+2}}. $$ This proves that the series $\sum_l \int_{E_l}|f(x)\psi(x)| \mathrm dx$ converges hence $\int_{\bigcup_l E_l}|f(x)\psi(x)|\mathrm dx $ converges.