Dominated Convergence Theorem

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $(X,\mathcal{F},\mu)$ be a measure space. Let $(f_{m})_{m\in\mathbb{N}}$ be a sequence in $\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$, that is, of measurable functions $f_{m}:X\to\mathbb{R}$, and let $f:X\to\mathbb{R}$ be a function with

$$f_{m}(x)\to f(x)\qquad\text{for every }x\in X .$$

Let $g\in\mathcal{L}^{1}(\mu)$ satisfy $|f_{m}(x)|\le g(x)$ for every $m\in\mathbb{N}$ and every $x\in X$.

(i) Integrability. $f\in\mathcal{L}^{1}(\mu)$, and $f_{m}\in\mathcal{L}^{1}(\mu)$ for every $m\in\mathbb{N}$.

(ii) Convergence in mean. $|f_{m}-f|\in\mathcal{L}^{1}(\mu)$ for every $m\in\mathbb{N}$, and, for the integrals of these elements of $\mathcal{L}^{1}(\mu)$,

$$\int_{X}|f_{m}-f|\,d\mu\to0 .$$

(iii) Convergence of the integrals.

$$\int_{X}f_{m}\,d\mu\to\int_{X}f\,d\mu .$$