Theorem (Dominated convergence): Let \(f_n\) be a sequence of measurable functions such that \(f_n \to f\). If there exists a measurable function \(g\) such that \(\abs{f_n(x)} \le g(x)\) for all \(n\) and all \(x\), then
\[\lim_{n \to \infty} \int f_n \, d\mu= \int f \, d\mu.\]Proof: Rudin, Theorem 11.32
A straightforward corollary of the DCT is that if \(\{f_n\}\) is uniformly bounded by a constant, then the DCT clearly applies.
Corollary (Bounded convergence): If \(\int d\mu < \infty\), \(\{f_n\}\) is uniformly bounded, and \(f_n \to f\), then
\[\lim_{n \to \infty} \int f_n \, d\mu= \int f \, d\mu.\]The theorem can also be stated in terms of expected values and convergence in distribution:
Theorem (Dominated convergence): If there exists a random variable \(Z\) such that \(\norm{\x_n} \le Z\) for all n and \(\Ex Z < \infty\), then \(\x_n \inD \x\) implies that \(\Ex \x_n \to \Ex \x\).