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'dominated convergence theorem'
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| Title of object: |
dominated convergence theorem |
| Canonical Name: |
DominatedConvergenceTheorem |
| Type: |
Theorem |
| Created on: |
2002-12-07 09:55:22 |
| Modified on: |
2004-10-15 18:15:34 |
| Classification: |
msc:28A20 |
| Synonyms: |
dominated convergence theorem=Lebesgue's dominated convergence theorem |
Revision comment (for changes between this and next version):
| Changes for correction #14665 ('Final statement'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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Content:
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.
A possible generalization is that if $\{f_r: r\in \mathbb{R}\}$ is a family
of measurable functions such that $|f_r|\leq |\Phi|$ for each $r\in \mathbb{R}$ and $f_r\xrightarrow[r\rightarrow 0]{} f$, then $f$ is integrable and
\[ \lim_{r\rightarrow 0} \int_X f_r = \int_X f. \] |
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