I think this is quite a simple question, but for some reason am finding it difficult to answer. The question is:
If $(f_n:n\in\mathbb{N})$ is a sequence of integrable functions, with $f_n \to f$ a.e. for some integrable $f$, then is it true that if $\|f_n\|_1 \to \|f\|_1 $, then $\|f_n-f\|_1\to 0$.
Any help is very good!
One way to prove this is to use the generalized dominated convergence theorem which states that if $|f_n(x)| \le g_n(x)$ ae., $f_n(x) \to f(x)$ ae., $g_n(x) \to g(x)$ ae., $f_n$ are measureable, $g_n,g$ are integrable and $\int g_n \to \int g$, then $\int f_n \to \int f$. (See https://math.stackexchange.com/a/88886/27978 for a proof.)
For this problem, let $g_n(x) = |f(x)|+|f_n(x)|$. Then $g_n(x) \to g(x)=2 |f(x)|$ ae., and by assumption, $\int g_n = \|f_n\|_1 + \|f\|_1\to 2 \|f\|_1= \int g$.
Since $|f(x) -f_n(x)| \le g_n(x)$ ae., and $(f(x) -f_n(x)) \to 0$ ae., we have $\|f-f_n\|_1 = \int |f-f_n| \to 0$.
