Let $\left(\Omega,\mathcal{F}\right)$ be some measurable space and let $P$ be a (probability) measure on $\Omega$. For another (probability) measure $Q$ such that $P>>Q$ the Kullback-Leibler divergence is defined as
$$ \operatorname{D_{KL}}(Q||P) = \int \ln \frac{dQ}{dP} dQ,$$
provided that this integral is well defined. Now if we denote the set of all $Q$ such that this expression exists and is finite by $M_P$, that is
$$M_P := \{Q \in \mathcal{M}(\Omega,\mathcal{F})| P>>Q, \operatorname{D_{KL}}(Q||P) < \infty \},$$
then what can we say about this set? Are there some known necessary and sufficient conditions on the measures?
According to this example it is e.g. not sufficient that there exists some measure $\mu$ that dominates both $P$ and $Q$.
Also, it would be interesting if we know something about the structure of $M_P$, for example, it should be convex: Since the function $x\ln \frac{x}{y}$ is convex in $x$, we would get $$Q,\hat{Q} \in M_P, \lambda \in [0,1] \implies \lambda Q + 1-\lambda \hat{Q} \in M_P.$$
Is there more we can say? Could it be a vector space (if we extend in a sensible way the definition to measures in general)?
The only thing that I found out so far is that it is not necessarily closed: One can consider measures $Q^n$ that converge in law to a point mass. Then of course the limit is not in $M_P$, but this comes more from the fact that it is not dominated anymore than from the actual quantity of interest here.