= Solution
The <monotone convergence theorem> states: on a <measure space> $(X,\mathcal F,\mu)$, let $u_n:X\to[0,\infty]$ be <measurable functions> with
$$
0\leq u_1(x)\leq u_2(x)\leq\cdots,\qquad u(x)=\lim_{n\to\infty}u_n(x).
$$
Then $u$ is measurable and
$$
\boxed{\int_Xu\,d\mu=\lim_{n\to\infty}\int_Xu_n\,d\mu.}
$$
The <integrals> and their <limit> may be infinite. No integrable dominator, <finite measure> hypothesis, or <sigma-finiteness> is required. The nonnegativity and increasing order are the essential hypotheses in this form of the theorem.
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