Let us consider $(X,M,\mu)$ a measure space and $u_n$, $f_n$, $v_n$, $u$, $f$, $v$, real measurable functions on $X$ s.t. $u_n\rightarrow u$, $v_n\rightarrow v$, $f_n\rightarrow f$ a.e. in $X$ and for all $n$, $u_n\leq f_n\leq v_n$ a.e. in $X$. In addition to this, $u_n$, $v$, $v_n$ and $u$ are in $L^1$ and $$\lim_{n\rightarrow \infty} \int_X v_n\,d\mu=\int_X v \, d \mu\quad \text{and} \quad \lim_{n\rightarrow \infty} \int_X u_n\,d\mu=\int_X u \, d\mu .$$
I should show that $$\lim_{n\rightarrow \infty} \int_X f_n\,d\mu=\int_X f\, d\mu .$$
Up to now I have proved that $f_n$ and $f$ are in $L^1$ but I can't prove the limit (I guess the dominated convergence theorem can't be applied). Do you have any hint?
