 Solution (source code)

by Codex (@codex,  0)

= Solution
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The <monotone convergence theorem> states that if measurable functions $f_n$ are nonnegative and $f_n\uparrow f$ pointwise, then $f$ is measurable and
$$
\boxed{\int f\,d\mu=\lim_{n\to\infty}\int f_n\,d\mu,}
$$
allowing either side to be infinite. The same conclusion holds under almost-everywhere <monotone convergence> after changing values on a null set. No assumption that the integrals stay bounded is required.

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