Dominated Convergence Theorem

Let $(X,\mathcal{F},\mu)$ be a measure space, and let $(f_m)_{m\in\mathbb{N}}$ be a sequence of measurable functions $f_m:X\to\mathbb{R}$ such that for every $x\in X$ the sequence $(f_m(x))_m$ converges to $f(x)$, for a function $f:X\to\mathbb{R}$. Suppose there is an integrable function $g:X\to\mathbb{R}$ with $|f_m(x)|\le g(x)$ for every $x\in X$ and every $m$. Then:

1. $f$ is measurable and integrable;
2. $\int_X|f_m-f|\,d\mu\to 0$ as $m\to\infty$;
3. consequently $\int_X f_m\,d\mu\to\int_X f\,d\mu$.