Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric\*
​$$ d(P,Q) = \frac12\max\_{\emptyset\neq A\subseteq X} \|P​(\cdot\mid A)-Q(\cdot\mid A)\|\_1. $$
Obviously, the total variation metric $\frac12\|P-Q\|\_1$ is majorized by $d(P,Q)$.


Question: has anyone encountered $d(P,Q)$ in the literature? Does it have a name?


\*It's not immediately obvious that $d$ satisfies the triangle inequality, but I think this can be shown.



A metric stronger than total variation
I think it is just called (scaled) **supremum $L\_1$ norm** and it is mostly studied in Bayesian nonparametric estimation literature, especially posterior consistency. The following paper investigate conditional density yet it is obvious that corresponding probability measure is exactly what you wrote down.



> 
> De Blasi, Pierpaolo, and Stephen G. Walker. "Posterior asymptotics in
>  the supremum $ L\_ {1} $ norm for conditional density estimation."
>  Electronic Journal of Statistics 10.2 (2016): 3219-3246.
> 
> 
> 


