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...