Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 23J / b / Solution

The dominated convergence theorem states that, on a measure space, if measurable $f_n\to f$ almost everywhere and $|f_n|\le g$ almost everywhere for one integrable $g$, then $f$ is integrable and $\int |f_n-f|\to 0$. In particular, $\int f_n\to \int f$. The statement applies to real or complex functions.