| Version 4 |
Version 3 |
| Given two signed measures $\mu$ and $\nu$ on the same measurable space |
Given two signed measures $\mu$ and $\nu$ on the same measurable space |
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$(\Omega, \mathscr{S})$, we say that $\nu$ is \emph{absolutely continuous}
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$(\Omega, \mathcal{A})$, we say that $\nu$ is \emph{absolutely continuous}
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with respect to $\mu$ if, for each $A\in \mathscr{S}$ such that $|\mu|(A)=0$,
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with respect to $\mu$ if, for each $A\in \mathcal{A}$ such that $|\mu|(A)=0$,
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| it holds $\nu(A)=0$. This is usually denoted by $\nu \ll \mu$. |
it holds $\nu(A)=0$. This is usually denoted by $\nu \ll \mu$. |
| \textbf{Remarks.} |
\textbf{Remarks.} |
| If $(\nu^+, \nu^-)$ is the Jordan decomposition of $\nu$, the following \PMlinkescapetext{propositions} are equivalent: |
If $(\nu^+, \nu^-)$ is the Jordan decomposition of $\nu$, the following \PMlinkescapetext{propositions} are equivalent: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\nu\ll\mu$; |
\item $\nu\ll\mu$; |
| \item $\nu^+\ll\mu$ and $\nu^-\ll\mu$; |
\item $\nu^+\ll\mu$ and $\nu^-\ll\mu$; |
| \item $|\nu|\ll\|\mu|$. |
\item $|\nu|\ll\|\mu|$. |
| \end{enumerate} |
\end{enumerate} |
| If $\nu$ is a finite signed measure and $\nu\ll\mu$, the following useful property holds: for each $\varepsilon>0$, there is a $\delta>0$ such that |
If $\nu$ is a finite signed measure and $\nu\ll\mu$, the following useful property holds: for each $\varepsilon>0$, there is a $\delta>0$ such that |
| $|\nu|(E)<\varepsilon$ whenever $|\mu|(E)<\delta$. |
$|\nu|(E)<\varepsilon$ whenever $|\mu|(E)<\delta$. |
