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Homedominated convergence theorem

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# dominated convergence theorem

Let $X$ be a measure space, and let $\Phi,f_{1},f_{2},\dots$ be measurable functions such that $\int_{X}\Phi<\infty$ and $|f_{n}|\leq\Phi$ for each $n$. If $f_{n}\rightarrow f$ almost everywhere, then $f$ is integrable and

$\lim_{{n\rightarrow\infty}}\int_{X}f_{n}=\int_{X}f.$ |

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

Related:

MonotoneConvergenceTheorem, FatousLemma, VitaliConvergenceTheorem

Synonym:

Lebesgue's dominated convergence theorem

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

28A20*no label found*

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