# Is a Cauchy distribution absolutely continuous with respect to a Gaussian distribution?

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.

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.