The Kullback-Leibler (KL) divergence (which I am familiar with) is usually defined for probability distributions. However, even though some people claim that a probability distribution is just a synonym for probability measure, that is, some people claim that a probability distribution is a function $\mathbb{P}$ from the event space $\mathcal{F}$ to $[0, 1]$, or, more formally, $\mathbb{P}: \mathcal{F} \rightarrow [0, 1]$, other people claim that the definition of a probability distribution depends on the context.
For example, the Wikipedia article on the KL divergence defines the KL divergence as follows
For discrete probability distributions $P$ and $Q$ defined on the same probability space, $\mathcal {X}$, the Kullback–Leibler divergence of $Q$ from $P$ is defined to be
$$ D_\text{KL}(P \parallel Q) = \sum_{x\in\mathcal{X}} P(x) \log\left(\frac{P(x)}{Q(x)}\right) $$
A probability space $\mathcal{X}$ is usually defined as a triple consisting of the sample space $\Omega$, the event space $\mathcal{F}$ and the probability measure $\mathbb{P}$. So, it seems that the definition above says that a probability distribution (or probability measure, assuming that a probability distribution is a synonym for probability measure) is defined on a probability space, which is composed of a probability measure, so it seems that it states that probability distribution is defined on a probability distribution, which makes little or no sense. So, in the definition above, $P$ and $Q$ should not be probability measures. What's going on?
Is the Kullback-Leibler divergence defined for probability measures or random variables, or something else (e.g. cumulative distribution functions)?
For two probability measures $\nu$ and $\mu$ s.t. $\nu\ll\mu$, the KL divergence of $\nu$ w.r.t. $\mu$ is defined as $$ D_{KL}(\nu\,||\,\mu)=\mathsf{E}^{\nu}\ln\left(\frac{d\nu}{d\mu}\right),\label{1}\tag{1} $$ where $d\nu/d\mu$ is the Radon-Nikodym derivative of $\nu$ w.r.t. $\mu$. If $\nu$ is not absolutely continuous w.r.t. $\mu$, $D_{KL}(\nu\,||\,\mu)=\infty$.
Indeed, a probability distribution is a probability measure (typically, on $\mathbb{R}^d$). However, when random variables are involved, their distributions are push-forward measures. In this case $D_{KL}$ "between random variables" is $\eqref{1}$ applied to the corresponding distribution functions.