# suppose $\{f_n\}$ is a sequence of nonnegative measurable defined on a measurable set $E$. Prove the following inequality.

$\int_E \liminf _{n \rightarrow \infty} f_n \leq \int_E \liminf _{n \rightarrow \infty} f_n$

This problem appears in a textbook. Is this a typo?

Also, I don't know whether the sequence of functions converges, or whether it is dominated by a function. How can this be solved?

• Look up "Fatou's inequality". – E. Joseph Nov 20 '16 at 14:42
• Yes, but don't I need the sequence to conv. ptwise a.e. to use Fatou's lemma? – Ninosław Brzostowiecki Nov 20 '16 at 14:50
• No, the limit inf is always well defined. – E. Joseph Nov 20 '16 at 14:54
• Both sides of the inequality are the same. – PhoemueX Nov 21 '16 at 21:49

You can define $g_n=inf\{f_k:k\ge n\}$ then every $g_n$ is nonnegative, $\{g_n\}$ is an increasing sequence and $\lim_n g_n=\lim \inf f_n$ as n goes to infinity and then you can apply the monotone convergence theorem.