I have this problem which is exactly the opposite of what I would find easy to prove! Consider a sequence of Lebesgue measurable non-negative functions $(f_n)$ such that $f_n\rightarrow f$ pointwise for some $f\in L^1$, and also that $\int_{\mathbb{R}}f_n\rightarrow \int_{\mathbb{R}}f$. Prove $||f_n-f||_1 \rightarrow 0$.
This is actually giving me a hard time because it goes against my intuition. How does pointwise convergence and integral convergence imply convergence in $L^1$? Why? How?
Any help would be appreciated... Thanks.
$\int (f-f_n)^{+} \to 0$ by Dominated Convergence Theorem since $(f-f_n)^{+} \to 0$ pointwise and $(f-f_n)^{+} \leq f$. Also $\int (f-f_n) \to 0$ implies $\int [(f-f_n)^{+} -(f-f_n)^{-}] \to 0$. Combining these two we get $\int (f-f_n)^{-} \to 0$. Now $\int |f-f_n| = \int (f-f_n)^{+} +\int (f-f_n)^{-} \to 0$