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 $\lbrace f_n\rbrace$ is uniformly bounded by a constant, then the DCT clearly applies.
Corollary (Bounded convergence): If $\int d\mu < \infty$, $\lbrace f_n\rbrace$ 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 $\Ex \x_n \to \Ex \x$.