Let $(X,\mathcal{A},\mu)$ be a measure space. let $(f_n)_n,$ $(h_n)_n$ be sequences in $L^1,$ such that $(f_n)_n$ converges in measure to $f \in L^1$ and $(h_n)_n$ converges in measure to $h \in L^1.$ Let $(\phi_n)_n$ be such that $\forall n \in \mathbb{N},f_n\leq \phi_n\leq h_n,(\phi_n)_n$ converges in measure to $\phi.$ We suppose also that $\lim_n \int_Xf_nd\mu=\int_Xfd\mu,\lim_n\int_X h_nd\mu=\int_Xhd\mu.$
1) Prove that $\lim_n\int_X\phi_nd\mu=\int_X\phi d\mu$
2) If also, $f_n \leq 0 \leq h_n,$ prove that $(\phi_n)_n$ converges to $\phi$ in $L^1$
When we have convergence a.e the problem is easy, using Fatou's lemma, but here we have convergence in measure, it seems also Fatou's lemma exist for convergence in measure, so if I proved that $\int_X|f|d\mu \leq\liminf_n \int_X|f_n|d\mu$ then the problem is solved (We apply it to $h_n-\phi_n \geq0$ and $\phi_n-f_n\geq 0$).
I appreciate if you can check my solution, to prove the Fatou's lemma for convergence in measure.
$$\forall x >0,\forall n,k \in \mathbb{N},\mu(\left\{ |f|>x+\frac{1}{k}\right\}) \leq \mu(\left\{ |f_n-f|>\frac{1}{k}\right\})+\mu(\left\{ |f_n|>x\right\})$$
So, $$\forall x >0,\forall k \in \mathbb{N},\mu(\left\{ |f|>x+\frac{1}{k}\right\}) \leq \liminf_n\mu(\left\{ |f_n|>x\right\})$$
then take the limit $k \to +\infty$
$$\forall x>0,\mu(\left\{ |f|>x\right\}) \leq \liminf_n\mu(\left\{ |f_n|>x\right\})$$
Then integrate taking $x>0,$ and using Fatou's lemma (which we know)
$$\int_0^{+\infty}\mu(\left\{ |f|>x\right\})dx \leq \liminf_n\int_0^{+\infty}\mu(\left\{ |f_n|>x\right\})dx$$
Conclusion $$\int_X|f|d\mu \leq \liminf_n\int_X |f_n|d\mu$$
For the part 2), we can apply Fatou's lemma to $|\phi|+h_n-f_n-|\phi_n-\phi|\geq0$ (In fact we can also apply Pratt's, as an application of part 1))