Lebesgue's Dominated Convergence Theorem

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\left({f_n}\right)_{n \in \N} \in \mathcal{L}^1 \left({\mu}\right)$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions.

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


 * $\forall n \in \N: \left\vert{f_n}\right\vert \le g$ pointwise

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

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


 * $\displaystyle \lim_{n \to \infty} \int \left\vert{f_n - f}\right\vert \, \mathrm d \mu = 0$
 * $\displaystyle \lim_{n \to \infty} \int f_n \, \mathrm d \mu = \int \lim_{n \to \infty} f_n \, \mathrm d \mu$