Is there an equivalent of KL divergence between probability measures that are not absolutely continuous with respect to the Lebesgue measure?
I am interested in rigorous notions of typical and atypical (rare) trajectories in a dynamical system. We can assume uniform hyperbolicity for simplicity and the existence and smoothness of an SRB measure. Then, I would like to say something like this: If I have a trajectory that behaves as if it was generated by a probability measure $\mu$, then, I would consider this trajectory $\epsilon$-atypical if there is a set $A \subset S$ such that $\mu < \epsilon$ and $\mu_{SRB} > 1 - \epsilon$.
It would be nice if there was a global definition of distance between $\mu$ and $\mu_{SRB}$ that would allow an extension to Sanov's theorem to make rigorous how likely it is to encounter a rare trajectory.
Thank you very much for your time!