Given three probability measures on $N$ elements (so $\mu_0, \mu_1,\mu_2 \in \ell^1_N$), I need to define the difference of $\mu_1$ and $\mu_2$ "modulo" $\mu_0$ as $$ \sup \bigg\{ \int f \mathrm{d}(\mu_1 - \mu_2) \;\bigg|\; \|f\|_\infty \leq 1 \text{ and } \int f \mathrm{d}\mu_0 =0 \bigg\}. $$ This is the [weak] norm of $\mu_1-\mu_2$ when one restricts to the annihilator of $\mu_0$.

Question: Is there any probabilistic or geometric interpretation of this quantity? Does it bears a name or has it been studied somewhere?

Given three probability measures on $N$ elements (so $\mu_0, \mu_1,\mu_2 \in \ell^1_N$), I need to define the difference of $\mu_1$ and $\mu_2$ "modulo" $\mu_0$ as $$ \sup \bigg\{ \int f \,\mathrm{d}(\mu_1 - \mu_2) \;\bigg|\; \|f\|_\infty \leq 1 \text{ and } \int f \,\mathrm{d}\mu_0 =0 \bigg\}. $$ This is the [weak] norm of $\mu_1-\mu_2$ when one restricts to the annihilator of $\mu_0$.

Question: Is there any probabilistic or geometric interpretation of this quantity? Does it bear a name or has it been studied somewhere?