dominated convergence theorem

Let $X$ be a measure space, and let $\mathrm{\Phi },f_1,f_2,\mathrm{\dots }$ be measurable functions such that and $|f_n|\le \mathrm{\Phi }$ for each $n$. If $f_n\to f$ almost everywhere, then $f$ is integrable and

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_X{f}_n={\int }_{X}f.$$

This theorem is a corollary of the Fatou-Lebesgue theorem.

Title	dominated convergence theorem
Canonical name	DominatedConvergenceTheorem
Date of creation	2013-03-22 13:12:47
Last modified on	2013-03-22 13:12:47
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	13
Author	Koro (127)
Entry type	Theorem
Classification	msc 28A20
Synonym	Lebesgue’s dominated convergence theorem
Related topic	MonotoneConvergenceTheorem
Related topic	FatousLemma
Related topic	VitaliConvergenceTheorem