dominated convergence theorem

Let $X$ be a measure space, and let $\mathrm{\Phi },f_1,f_2,\mathrm{\dots }$ be measurable functions such that and $|f_n|\le \mathrm{\Phi }$ for each $n$.If $f_n\to f$ almost everywhere, then $f$ is integrable and

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_X{f}_n={\int }_{X}f.$$

This theorem is a corollary of the Fatou-Lebesgue theorem.