In an article I'm currently reading, a reasoning is used that I don't understand.
We have an integral of a function over a domain with both depending on the same $\epsilon>0$. They show that $$\displaystyle\int_{D_\epsilon}f_\epsilon g_\epsilon dx \to \displaystyle\int_{D}f_0 g_0 dx (*)$$
as $\epsilon\to 0$ (after passing to a subsequence/subfamily). For this they show that $f_\epsilon \to f_0$ in $L^p(D)$ and $g_\epsilon$ is bounded in $L^{p'}(D)$ where $\frac{1}{p}+\frac{1}{p'}=1$
Also we have that $\mathcal{L}^n(D\setminus D_\epsilon)$ vanishes. Then they say that a standard argument yields the claim. However I don't understand this. The thing missing for me is some convergence of the $g_\epsilon$ term. I have tried manipulating the first term above with some artificial zeros but something is missing. I would be happy about every help, thanks!
