For $X$ a measurable space and $P,Q$ two probability measure on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dP,$$ where $\frac{dQ}{dP}$ is a Radon-Nikodym derivative of $Q$ with respect to $P$. Now, the question is,

What is the sufficient condition for $D(Q\|P)=0$ when $D(Q\|P)$ in defined in infinite probability space?

By infinite probability space I mean "sample space, $\Omega$, which is the set of all possible outcomes, is not finite."