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 $$\displaystyle\int_{D_\epsilon}f_\epsilon\,dx \to \displaystyle\int_{D}f\,dx \,\,\,\,\,\,\,\, (*)$$
as $\epsilon\to 0.$ There the integrand $f_\epsilon$ converges a.e. to $f$ and $\mathcal{L}^n(D_\epsilon\setminus D)$ vanishes. First they show of course that limit and integration can be changed in the first term. So far so good. Then however they say, that $(*)$ only holds for a subsequence and not the whole sequence. Of course it doesn't matter if we have to pass to a subsequence but I can't understand why that is? I thought about using the fact that $L^1$ convergence implies existence of a subsequence with convergence a.e. but I am not sure whether this is used or not here. Thanks in advance!
