= Solution
The <dominated convergence theorem> states: on a <measure space> $(X,\mathcal F,\mu)$, let $u_n$ be <measurable functions> converging <almost everywhere> to a measurable $u$. Suppose there is a nonnegative <measurable function> $G$ with $\int_XG\,d\mu<\infty$ and $|u_n|\leq G$ <almost everywhere> for every $n$. Then $u$ is integrable and
$$
\boxed{\int_X|u_n-u|\,d\mu\longrightarrow0,\qquad
\int_Xu_n\,d\mu\longrightarrow\int_Xu\,d\mu.}
$$
Here integrability means integrability with respect to the given <measure>; for <Lebesgue measure> it is <Lebesgue integrability>. Neither a <finite measure> space nor monotonicity of the sequence is required. The same single integrable dominator must control the whole sequence.
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