From Donald Cohn's Measure Theory, section 2.4, exercise 10.
Let $(X, A, \mu)$ be a measure space, and let $f$ and $f_1, f_2, \dots$ be non-negative functions that belong to $L^1(X, A, \mu, R)$ and satisfy
(i) $\{f_n\}_n$ converges to $f$ almost everywhere;
(ii) $\int fd\mu = \lim_n\int f_nd\mu$.
Show that $\lim_n\int |f_n – f|d\mu = 0$.
I let $f_n = 1/n$ on $[n, n+1]$ and $0$ elsewhere, and let $f=0$, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed.
Best Answer
Hint:
Fatou's Lemma yields $$ 2\int f = \int \liminf_{n\to\infty} \left ( f_n + f - |f_n - f| \right ) \le \int f + \int f + \liminf_{n\to\infty} \left ( - \int |f_n - f| \right ). $$