TITLE: Relative entropy between singular measures
QUESTION [7 upvotes]: Usually, to define relative entropy between two probability measures, one assumes absolute continuity. Is it possible to extend the usual definition in the non absolutely continuous case?

REPLY [3 votes]: Relative entropy between two probability measures $P$ and $Q$ can be defined even if $P$ is not absolutely continuous w.r.to $Q$. In any case, $P$ and $Q$ are absolutely continuous w.r.to a common measure $\mu$ (one can take $\mu$ to be $\frac{P+Q}{2}$). Then relative entropy between $P$ and $Q$ is defined as $$D(P\|Q)=\int p\log\frac{p}{q}d\mu,$$ where $p=dP/d\mu$ and $q=dQ/d\mu$.