When reading about Sobolev Spaces, some results are proven using the density of smooth function.
So they prove the results on smooth function and then conclude by taking limits within the integral.
For instance take $(f,g) \in W^{1,p}(\mathbb{R}^d) \times W^{1,p'}(\mathbb{R}^d) , (f_n, g_n)$ two sequences of smooth function approaching $f$ and $g$. To prove that $fg \in W^{1,1}(\mathbb{R}^d) \ and \ \nabla fg \ = \ f\nabla g \ + \ g\nabla f$ we can easily see that $fg$ and the RHS are in $L^1(\mathbb{R}^d)$ thanks to Holder inequality and integrating by parts that $\int_{\mathbb{R}^d}f_n g_n \delta_{i}\varphi \ = \ -\int_{\mathbb{R}^d}f_n \delta_{i}g_n \varphi \ -\int_{\mathbb{R}^d}\delta_{i}f_n g_n \varphi $ and then we take the limit to prove the results.
Here I can see that $f_n g_n $ converges to $fg $ in $L^1$ so $f_n g_n \delta_{i}\varphi$ converges to $fg \delta_{i}\varphi$ in $L^1$ so $\int_{\mathbb{R}^d}f_n g_n \delta_{i}\varphi$ converges to $\int_{\mathbb{R}^d}fg \delta_{i}\varphi$
However taking the limit within the integral doesn't seem that trivial for me in every proof I see. Is there a general result I missed linking convergence in $W^{m,p} \ or \ L^p $ to convergence in integral ?
Thanks
It is a general result that smooth functions are dense in Sobolev spaces, i.e. for any $f\in W^{k,p}(\mathbb{R^d})$, one can find a sequence $(f_n)_{n\in\mathbb{N}}\subset C^\infty(\mathbb{R}^d)$, so that $f_n\longrightarrow f$ with respect to the norm $\|\cdot\|_{W^{k,p}(\mathbb{R}^d)}$.
E.g. for $k=1$, this norm is given by $$\|f\|_{W^{k,p} (\mathbb{R}^d)} = \|f\|_{L^p(\mathbb{R}^d)} + \sum\limits_{i=1}^d\|\partial_i f\|_{L^p(\mathbb{R}^d)},$$
and so one gets $f_n\longrightarrow f$, as well as $\partial_i f_n \longrightarrow \partial_i f$ with respect to $\|\cdot\|_{L^p(\mathbb{R}^d)}$.
Applied to your case, since $f,\partial_i f\in L^{p}(\mathbb{R}^d)$ and $g,\partial_i g \in L^{p'}(\mathbb{R}^d)$, one can use Hölder's inequality to show that $f_n(\partial_i g_n) \longrightarrow f(\partial_i g)$ and $(\partial_i f_n) g_n \longrightarrow (\partial_i f) g$ in $L^1(\mathbb{R}^d)$, so that the limit can be taken within the integral.