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'$\sigma$-finite'
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| Title of object: |
$\sigma$-finite |
| Canonical Name: |
SigmaFinite |
| Type: |
Definition |
| Created on: |
2002-02-27 00:29:13 |
| Modified on: |
2003-07-15 15:32:44 |
| Classification: |
msc:28A10 |
| Defines: |
$\sigma$-infinite, sigma-infinite |
| Synonyms: |
$\sigma$-finite=$\sigma$ finite
$\sigma$-finite=sigma-finite
$\sigma$-finite=sigma finite |
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Content:
\paragraph{Definition}
Let $(\Omega, \mu)$ be a measurable space. Let $\sequence{A_n}$ be a finite or countable sequence such that $\mu(A_k) < \infty$ and $\Omega = \cup_{n} A_n$. $\mu$ is called \emph{$\sigma$-finite} (or \emph{sigma-finite}) iff
\begin{equation*}
\mu(\Omega) < \infty.
\end{equation*}
If $\mu$ is not $\sigma$-finite, it is called \emph{$\sigma$-infinite}. |
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