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."

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as $$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$ 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."

What is the sufficient condition for relative entropy to be zero when we define the relative entropy in infinite probability space?

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."