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Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says that if $P$ is absolutely continuous with respect to $Q$, then there exists a finite measurable function $f$ on $\Omega$ that satisfies

$$ P(A) = \int_A f dQ $$

If $P$ is fixed and we consider a change in the function $f$, what does this imply about $Q$? Are we changing the probability measure with respect to which $f$ is the Radon-Nikodym derivative of $P$?

For instance, in Girsanov theory one can choose $f$ to move from a possibly more complex probability measure $Q$ to a simpler one $P$, but what is the intution for the effect of some change in $f$ on this new measure $P$?

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    $\begingroup$ The Radon-Nikodym theory says that: if $P$ is absolutely continuous with respect to $Q$ ($P \ll Q$) then there exists a finite measurable function $f$ on $\Omega$ that satisfies $$ P(A) = \int_A f dQ $$ $\endgroup$ – Ramiro May 3 '16 at 16:53

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