Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $Q$ be the uniform distribution on $(\Omega, \mathcal{F})$ such that $q = dQ / d\mu$ exists. Then the KL-divergence for some probability measure $P$ with $p = dP / \mu$ is given by
$$
D\_{\text{KL}}(P || Q) = \int\_\Omega p \log \frac{p}{q} d\mu.
$$i.e. we view the KL-divergence as a function of only $P$. If $\Omega$ is finite then I believe that the KL-divergence is bounded above by $|\Omega|\log|\Omega|$. But what if, more generally, $\Omega$ is compact and $\mathcal{F}$ is the Borel $\sigma$-algebra?



Upper bound for KL divergence on compact space
First, the definition of the KL divergence has nothing to do with compactness as it is defined entirely in terms of the density of one measure with respect to the other one.


Second, the KL divergence is **not** bounded even for finite spaces. Take $\Omega$ to be a two point set, $P$ be the distribution with the weights $(1/2,1/2)$, and $Q$ be $(\epsilon, 1-\epsilon)$ with $\epsilon$ close to 0.


