a homework question from measure and integraiton theory course.
Suppose $f_n \in L_1(\mathbb R^d)$ for each $n\in\mathbb N$， $f_n\geq 0$ ， $f_n\to f$ a.e and $\int f_n\to\int f<\infty$.
Prove that $\int|f_n-f| \to 0$
(Hint: ($f_n - f)_-\leq f$. Use dominated convergence theorem )
I am thinking that since $|f_n -f | \to 0$ a.e and $|f_n-f|=(f_n - f)_+ + (fn-f)_-$ . If I can show $(f_n-f)_+ ≤ f$ and $(f_n-f)_-≤f$ . Then since $f_n$ and $f$ are integrable , by DCT, $\int|f_n-f| \to \int0 =0$ . I think my approach might be wrong ..