Prove that $f\in L^{1}\left(\Omega\right)$ and $\lim_{n\rightarrow\infty}\int(\left|f_{n}\right|-\left|f_{n}-f\right|)=\int\left|f\right| $.

Question

Let $(f_n)$ be a sequence in $L^1(\Omega)$ such that:

(i) $f_n(x) \rightarrow f(x)$ a.e.

(ii) $(f_n)$ is bounded in $L^1(\Omega)$ i.e. $\left\Vert f_{n}\right\Vert _{L^{1}}\leq M\forall n $

Prove that $f\in L^{1}\left(\Omega\right)$ and $\lim_{n\rightarrow\infty}\int(\left|f_{n}\right|-\left|f_{n}-f\right|)=\int\left|f\right| $.

My previous post maybe usefull I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $.

Answer 1

Use my previous question I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $. with $a=f_n-f$, $b=f$ and consider the sequence $t_{n}=\left|\left|f_{n}\right|-\left|f_{n}-f\right|-\left|f\right|\right|$