Am I missing something in the following argument?
Here's the problem.
Let $(X, \mathscr{A}, \mu)$ be a measure space, and let $f,f_n$ be nonnegative functions that belong to $\mathscr{L}^1(X, \mathscr{A}, \mu, \mathbb{R})$. Prove that if $f_n \to f$ $\mu$-almost everywhere and $\int f_n \to \int f$, then $\int |f_n - f| \to 0$.
Notice that $|f_n - f| \leq f_n + f \in \mathscr{L}^1$. Also, $|f_n - f| \to 0$ almost everywhere. Then the dominated convergence theorem implies $\int |f_n - f| \to \int 0 = 0$.
Many arguments I've seen use Fatou or some other argument, but why isn't it as simple as the argument above?
In the dominated convergence theorem, you'd need that $|f_n - f| \leq g$, where $g$ has no dependence on $n$, is nonnegative, and integrable.