Category:Lebesgue's Dominated Convergence Theorem

This category contains pages concerning Lebesgue's Dominated Convergence Theorem:

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

Let $f : X \to \overline \R$ be a $\Sigma$-measurable function.

Let $g : X \to \overline \R_{\ge 0}$ be a $\mu$-integrable function.

Let $\sequence {f_n}_{n \mathop \in \N}$ be an sequence of $\Sigma$-measurable function $f_n : X \to \overline \R$ such that:

$\ds \map f x = \lim_{n \mathop \to \infty} \map {f_n} x$

and:

$\ds \size {\map {f_n} x} \le \map g x$

hold for $\mu$-almost all $x \in X$.

Then:

$f$ is $\mu$-integrable and $f_n$ is $\mu$-integrable for each $n \in \N$

and:

$\ds \int f \rd \mu = \lim_{n \mathop \to \infty} \int f_n \rd \mu$

Source of Name

This entry was named for Henri Léon Lebesgue.