Confusion about Fourier transform and fractional Sobolev space definition

Question

Consider the fractional Sobolev spaces defined as $$ W^{s,p}\left(\mathbb{R}\right):=\left\{ u\in L^{p}\left(\mathbb{R}\right):\int_{\mathbb{R}}\left(1+\left|\xi\right|^{sp}\right)\left|\widehat{u}\left(\xi\right)\right|^{p}d\xi<+\infty\right\} $$ where $s>0$, $1\leq p\leq+\infty$ and $\widehat{u}$ is the Fourier transform of $u$. From what I understand, these spaces, known also as Bessel potential spaces, are well-known and studied, but I have some trouble to understand the definition. The Fourier transform of $u\in L^{p}\left(\mathbb{R}\right)$ is, in general, a distribution, defined as $$\widehat{u}\left[\varphi\right]=\int_{\mathbb{R}}u\left(x\right)\widehat{\varphi}\left(x\right)dx$$ where $\varphi$ is a suitable function, for example $\varphi$ belongs to the Schwarz space. So, what is the meaning of $$\int_{\mathbb{R}}\left(1+\left|\xi\right|^{sp}\right)\left|\widehat{u}\left(\xi\right)\right|^{p}d\xi\,\,?$$ More precisely, how can we integrate a distribution written as a funciton of $\xi$? Maybe it is a simple question but I cannot find an answer in the text I studied.