I'm reading about tempered distributions in Folland's Real Analysis. In particular, tempered distributions are defined as the class of linear functionals on a Schwarz space. On page 293, Folland says:
If $f\in L_{\mathrm{loc} }^1$ and $\int (1+|x|)^N|f(x)|<\infty$ for some $N\ge 0$, then $f$ represents a tempered distribution since $\int |f \phi|\le C||\phi||_{(0,N)}$.
This statement is a bit puzzling. If $\int (1+|x|)^N|f(x)|<\infty$ for some $N$, then $\int |f(x)|<\infty$. So why bother looking at $N$ other than 0? I though perhaps "for some $N$" needed to be "for all $N$" but that doesn't seem necessary according to my understanding of convergence in the Schwarz space.
In fact, by choosing large $N$, it seems we strengthen our assumption and weaken our conclusion. What am I missing?
As pointed out by Daniel Fischer, the correct assumption should be $$\int (1+\lvert x\rvert)^{-N} \lvert f(x)\rvert < \infty$$ for some $N\geqslant 0$.