I read the proof of the following lemma and understand it completely. However the proof of the second part of lemma quite unclear to me. The author consider the case when $f(x)=0$ and I understood his reasoning but I am not able to transfer this when $f(x)=0$ a.e. and in some sense I know that the property of a.e. does not change but I would like to see the rigorous and detailed proof.
Frankly speaking, I have spent many hours in order to prove it by myself but no results. So would be very grateful for help!
If $f=0,$ then $\phi_n\to 0$ uniformly on $A_{\epsilon}$, so there is an integer $N$ such that if $n>N$ then $|\phi_n|<\epsilon$ on $A_{\epsilon}$. Calculating as above, we find
$|I_n|\le \int_E|\phi_n|=\int_{A_{\epsilon}}|\phi_n|+\int_{E-A_{\epsilon}}|\phi_n|\le \epsilon m(E)+Mm(E-A_{\epsilon})\le \epsilon m(E)+M\epsilon$ as soon as $n>N.$