Dominated Convergence Theorem

Let $(\Omega, \mathcal{F}, \mu)$ be a measure space (Definition~Measure Space). Suppose $f_n \to f$ pointwise $\mu$-a.e.\ and $|f_n| \leq g$ $\mu$-a.e.\ for all $n$, where $g \in L^1(\mu)$. Then $f \in L^1(\mu)$ and \[ \lim_{n \to \infty} \int f_n\, d\mu = \int f\, d\mu. \]