Suppose $f_n$ is as sequence of functions in $L^1[0,1]$ such that $f_n$ converges pointwise a.e. to $f\in L^1[0,1]$. Suppose also that $\int \vert f_n\vert \rightarrow \int \vert f\vert$. Is it true that $f_n$ converges to $f$ in the $L^1$ norm?
From Javaman's comment below: $|f_n-f|\leqslant |f_n|+|f|$. So DCT applies. Since $f_n\rightarrow f$ a.e. we have $\lim_n\int |f_n-f|=0.$ i.e $\Vert f_n-f\Vert \rightarrow 0.$
Yes you are right. Fatou's lemma can be applied here. As Javaman said, $|f_n-f|\le |f_n|+|f|$ is the key. Then $|f_n|+|f|-|f_n-f|\ge 0$, so Fatou's lemma applies. That is, $$\liminf_{n\to \infty}\int \left(|f_n|+|f|-|f_n-f|\right)\ge \int \liminf_{n\to \infty} \left(|f_n|+|f|-|f_n-f|\right)$$
Hence $$2\int |f|-\limsup_{n\to \infty}\int|f_n-f|\ge 2\int |f|$$
Thus $$\limsup_{n\to \infty}\int|f_n-f|\le 0$$ and we are done.
For reference, I just wanted to add that one may use the generalized dominated convergence theorem here.
Generalized Dominated Convergence Theorem: Let $(X,\mathscr{M},\mu)$ be a measure space. Suppose $\{f_n\},\{g_n\}\subseteq L^1(\mu)$, $f,g\in L^1(\mu)$ with $f_n\to f$ and $g_n\to g$ almost everywhere, $|f_n|\leq g_n$, and $\int{g_n\,\mathrm{d}\mu}\to\int{g\,\mathrm{d}\mu}$. Then $\int{f_n\,\mathrm{d}\mu}\to\int{f\,\mathrm{d}\mu}$.
In our case, we may note that $|f_n-f|\leq|f_n|+|f|\in L^1$ for every $n$. Of course, $|f_n-f|\to0\in L^1$ and $|f_n|+|f|\to2|f|\in L^1$ almost everywhere. Moreover. $\int{|f_n|+|f|}\to\int{2|f|}$ by assumption. Hence, the conclusion follows by the generalized dominated convergence theorem.
