Convergence in $L^1(0,1)$

Question

Let $f_k$, $f \in L^1(0,1)$ with $f_k, f \geq 0$ in $(0,1)$. Suppose $f_k\rightarrow f$ a.e. and $$\int_0^1 f_k \rightarrow \int_0^1 f \, . $$ How could one prove that $f_k \rightarrow f$ in $L^1(0,1)$ ?

Answer 1

Define $g_k^+=\max\{f_k-f,0\}$ and $g_k^-=-\min\{f_k-f,0\}$. Then $0\leq g_k^-\leq f$ is dominated and converges point-wise to zero. By dominated convergence $\lim_k \int g_k^- =0$. Now $$ \int g_k^+ = \int (f_k-f) - \int g_k^- $$ and the RHS goes to zero. Finally $\|f_k-f\|_1=\int g_k^+ + \int g_k^-$.