Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous function $g$ a.e.
I showed (by Holder) that for all $f \in H^s$, and $s >d/2$, then $\hat{f} \in L^1({\mathbb R}^d)$. I want to show that there is a continuous function $g$ such that $f=g$ a.e. I argued this way but confused in the last step: Let $x \in \mathbb{R}^d$, and fix $\epsilon >0$, then for $s >d/2$. Since there is $g \in C_c$ with $supp(g) \subset K \subset \subset \mathbb{R}^d$ and $\|g-f\|_{L^2}<\epsilon$. Then
$$|f(x)-g(x)| \leq \int (1+|x|^2)^{-s/2} (1+|x|^2)^{-s/2}|f(x)-g(x)|dx$$
I applied Holder and used the properties to close $g$ to prove the terms are small except the term
$$\int_{K^c} (1+|x|^2)^s |f(x)|^2 dx$$
It is clearly bounded by $\|f\|_{H^s} =\|\hat{f}\|_{H^s}$, but how cal I prove it is small, i.e. how can I use Riemann-Lebesgue lemma? comment is enough. Thanks