Lebesgue's Dominated Convergence Theorem

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Theorem

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

Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\mathcal L^1} \mu$, $f_n: X \to \R$ 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:

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


Proof


Source of Name

This entry was named for Henri Léon Lebesgue.


Sources