Take a measure space $(\Omega, \mathcal{A}, \tau, P)$, with $P$ being a probability measure, and $\tau$ being a topology on $\Omega$.
I would like to construct a measure $Q$ such that $Q$ is equivalent to $P$, for its RN derivative exists, and that the RN derivative $f$ is as close as possible to $P$ itself as possible.
A first idea (from the RN derivatives definition) would be to define $Q$ via a neighbourhood filter $N_\omega \subset \tau$ of a point $\omega \in \Omega$ (similar to how one defines well defined conditional probability). Thus I would write down the expression
$Q(E) := \int\limits_E \lim\limits_{\{\omega\} \subset N_\omega} P(N_\omega) \; dP(\omega) $
Is this a sensible approach ? Is there any neat way of evaluating this expression ? Or is there an alternative approach which is more sensible ?
Thanks a lot in advance ! :)