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This stack exchange answer showed that Kullback-Leibler divergence between a Cauchy distribution and a Gaussian distribution is infinite.

Formally, $$KL(P||Q)=\infty$$, where $P$ is a Cauchy distribution with probability density function $$p(x)= \frac{1}{\pi}\frac{1}{1+x^2}$$ and $Q$ is a Gaussian distribution with pdf $$ q(x) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right) $$, both of which are defined over a real line.

My question is whether $P$ is absolutely continuous with respect to $Q$.

The reason why I am asking this is I want to know if the lack of absolute continuity is responsible for the infinite KL divergence.

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Yes, each is absolutely continuous with respect to the other. If, for instance, $\int_A p(x)\,dx = 0$ then you know $\int_A q(x)\,dx = 0$. And conversely.

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