= Solution
<Fatou's lemma> states that for nonnegative measurable $h_n$,
$$
\boxed{\int\liminf_n h_n\,d\mu\le\liminf_n\int h_n\,d\mu.}
$$
Indeed $g_n=\inf_{k\ge n}h_k$ is measurable, increases to $\liminf h_n$, and satisfies $g_n\le h_k$ for every $k\ge n$. The <monotone convergence theorem> gives $\int\liminf h_n=\lim_n\int g_n\le\lim_n\inf_{k\ge n}\int h_k$, proving the claim, including infinite values.
The <dominated convergence theorem> states that if measurable $h_n\to h$ <almost everywhere> and $|h_n|\le g$ <almost everywhere> for one integrable nonnegative $g$, then $h$ is integrable and $\int h_n\to\int h$; in fact $\int|h_n-h|\to0$. Integrability follows from $|h|\le g$. For real functions, Fatou applied to $g+h_n$ gives $\int h\le\liminf\int h_n$, and applied to $g-h_n$ gives the reverse inequality against the upper limit. Thus the integrals converge. To obtain the stronger conclusion, Fatou applied to $2g-|h_n-h|\ge0$ gives $2\int g\le2\int g-\limsup\int|h_n-h|$, forcing the latter limit to be zero. Complex functions may be treated through their real and imaginary parts or the same absolute-difference argument.
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