Lebesgue's Dominated Convergence Theorem

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^1} \mu$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions.

Suppose that for some $\mu$-integrable $g: X \to \R$, it holds that:


 * $\forall n \in \N: \size {f_n} \le g$ pointwise

Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists almost everywhere.

Then $f$ is $\mu$-integrable, and:


 * $\displaystyle \lim_{n \mathop \to \infty} \int \size {f_n - f} \rd \mu = 0$
 * $\displaystyle \lim_{n \mathop \to \infty} \int f_n \rd \mu = \int \lim_{n \mathop \to \infty} f_n \rd \mu$