Past exam of the options II course of the University of Oxford / 2010 / D / D3 / 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 $|f_n|\le g$ almost everywhere for some integrable $g$, then $f$ is integrable and $\int f_n\to \int f$. In fact $\int |f_n-f|\to 0$ as well. Both statements apply on any measure space.