Is there any ** official** (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?

Take, for example, the total variation distance: $$TV(\mu,\nu) = \sup_{A\in\mathcal{F}}|\mu(A)-\nu(A)|.$$

If $X$ and $Y$ are two real positive continuous random variables with densities $f_{X}$ and $f_{Y}$, then their total variation distance is, if I understand correctly: $$TV(\mu_{X},\mu_{Y}) = \int_{0}^{\infty} |f_{X}(z) - f_{Y}(z)|dz.$$

Would it make any sense to calculate a quantity, for $\tau > 0$, let's call it partial distance, like this: $$PV(\mu_{X},\mu_{Y};\tau) = \int_{\tau}^{\infty} |f_{X}(z) - f_{Y}(z)|dz\;\;\;?$$

If this does not make any sense (sorry, I really cannot tell, as I am not that good with measure theory...), can anyone think of a measure that would make sense?

What I want to use this for is to compare the closeness of two PDFs (or other functions describing a distribution: CDF, CCDF...) $f_{X}(t)$, $f_{Y}(t)$ to a third one $f_{Z}(t)$. I know that both $f_{X}$ and $f_{Y}$ "eventually" ($t\to\infty$) converge to $f_{Z}$, but I would like to show that one of them gets closer, sooner than the other one...

EDIT: I guess the underlying question for measure-theory people is: do these distance metrics (like total variation, Kullback-Leibler etc) really *need* to span all the elements of the sample space of the probability measure and if so, why?