= Solution
The <monotone convergence theorem> says that if $0\le f_n\uparrow f$ <almost everywhere> and the functions are <measurable> on a <measure space>, then
$$
\int f=\lim_n\int f_n,
$$
with infinite <integrals> permitted. The limit is <measurable> after the usual modification on any exceptional <null set>.
The <dominated convergence theorem> says that if <measurable functions> $f_n$ converge <almost everywhere> to $f$ and satisfy $|f_n|\le g$ <almost everywhere> for an integrable <nonnegative function> $g$, then $f$ is integrable and
$$
\boxed{\int|f_n-f|\longrightarrow0,\qquad\int f_n\longrightarrow\int f.}
$$
The theorem applies equally to real or complex functions. In particular domination must be by one fixed <Lebesgue integrable function>, rather than by a different bound for each $n$.
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