Take a measure space $(\Omega, \mathcal{A}, P)$, with $P$ being a probability measure.
I want to construct a second measure from $P$ via $Q(E) := \int_E P(\omega) \; dP(\omega)$. In other words, I would like to construct a probability measure $Q$, which is equivalent to $P$, such that the Radon Nikodym derivative of $P$ w.r.t. $Q$ remains $P$ (or as close as possible), in a sense that we would integrate $P$ as $P(\omega) = P(\{\omega\})$, to view $P$ as a measurable function on $\Omega$.
Does anyone have any ideas ? Thanks in advance :)