= Solution
<Fatou's lemma> states that nonnegative <measurable functions> satisfy
$$
\boxed{\int\liminf_nf_n\,d\mu\le\liminf_n\int f_n\,d\mu}.
$$
To prove it, put $h_n=\inf_{k\ge n}f_k$. These measurable nonnegative functions increase to $\liminf f_n$. By the <monotone convergence theorem>, $\int\liminf f_n=\lim_n\int h_n$. Since $h_n\le f_k$ for every $k\ge n$, one has $\int h_n\le\inf_{k\ge n}\int f_k$. Taking the limit proves the inequality, including extended infinite integrals.
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