Solution
= Solution
The <monotone convergence theorem> states that for nonnegative measurable functions $f_n\uparrow f$ almost everywhere, $\int f=\lim_n\int f_n$, allowing the value $+\infty$.
The <dominated convergence theorem> states that if measurable functions $f_n\to f$ almost everywhere and $|f_n|\leq g$ almost everywhere for some integrable $g$, then $f$ is integrable and $\int f_n\to\int f$. In fact $\int|f_n-f|\to0$ as well. Both statements apply on any <measure space>.