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Revision difference : $\sigma$-finite |
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Version 2 |
| \paragraph{Definition} |
\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 |
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*} |
\begin{equation*} |
| \mu(\Omega) < \infty. |
\mu(\Omega) < \infty. |
| \end{equation*} |
\end{equation*} |
| If $\mu$ is not $\sigma$-finite, it is called \emph{$\sigma$-infinite}. |
If $\mu$ is not $\sigma$-finite, it is called \emph{$\sigma$-infinite}. |
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