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2
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'Fatou-Lebesgue theorem'
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| Title of object: |
Fatou-Lebesgue theorem |
| Canonical Name: |
FatouLebesgueTheorem |
| Type: |
Theorem |
| Created on: |
2002-12-07 10:48:41.974716-05 |
| Modified on: |
2002-12-07 11:09:34.862891-05 |
| Classification: |
msc:28A20 |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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Content:
Let $X$ be a measure space. If $\Phi$ is a measurable function with
$\int_X \Phi< \infty$, and if $f_1, f_2,\dots$ is a sequence of measurable functions such that $|f_n|\leq \Phi$ for each $n$, then
\[g=\liminf_{n\rightarrow\infty} f_n \;\;\textnormal{and}\;
h=\limsup_{n\rightarrow\infty} f_n\]
are both integrable, and
\[-\infty < \int_X g \leq \liminf_{n\rightarrow\infty}\inf_X f_n \leq
\limsup_{k\rightarrow\infty}\int_X f_n \leq \int_X h < \infty.\] |
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