I have the following problem:
Let $(\Omega,F,\mu)$ be a measure space and $(f_n)_{n\in\mathbb{N}}$ be a sequence of non-negative integrable functions, so that $\lim_{n\rightarrow \infty} \int f_n d\mu$ exists. We define $f:=\lim f_n$, which exists. Show f is integrable and $$\lim_{n\rightarrow\infty} \int |f_n-f|d\mu=\lim_{n\rightarrow\infty}\int f_n d\mu-\int f d\mu$$
This reminds me of the dominated convergence theorem. But this theorem needs an integrable function $g$ with $|f(x)|\leq g(x)$. Why don't we need this here? Is this even connected to this theorem? Thanks in advance.
You can first prove that $\int f < \infty$ by using Fatou's lemma on the nonnegative functions $f_n$.
You can then prove the desired inequality in two stages:
1) Prove $\liminf_{n\rightarrow\infty} \int |f_n-f| \geq \lim_{n\rightarrow\infty} \int f_n - \int f$
2) Define $g_n(x) = \inf_{m\geq n} f_m(x)$, note that $g_n(x)\nearrow f(x)$, and use the monotone convergence theorem to prove the remaining inequality $$\limsup_{n\rightarrow\infty}\int |f_n-f| \leq \lim_{n\rightarrow\infty} \int f_n - \int f$$