Statistical Distance that doesn't require absolute continuity

Question

I was going through Kullback–Leibler divergence and it was mentioned in the wiki article that it requires absolute continuity. It was written that The Kullback–Leibler divergence is defined only if for all $x$, $Q ( x ) = 0$ implies $P(x)=0$ (absolute continuity). However, I am trying to find a statistical distance that doesn't follow this type of conditions. Thus, $Q(x)$ can be $0$ even though $P(x)$ is not. Is there any type of statistical distance possible without that condition?