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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?

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  • $\begingroup$ For example when $\mu_0$ is equidistribution (i.e. $\mu_0 = (\frac{1}{N},\frac{1}{N},...,\frac{1}{N})$), then the above sup is just $\|\mu_1-\mu_2\|_{\ell^1}$. $\endgroup$
    – ARG
    Apr 8, 2019 at 18:24
  • $\begingroup$ I just realised I should probably add an absolute value in the the sup (just to make sure this is really a norm). $\endgroup$
    – ARG
    Apr 8, 2019 at 20:02

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By using the standard duality, this quantity can be characterized (for an arbitrary triple of probability measures on the same space which need not be finite or countable) as $$ \min_{t\in\mathbb R} \| \mu_1 - \mu_2 - t\mu_0 \| \;, $$ where $\|\mu\|$ denotes the usual total variation of a signed measure. Although this definition seems to be quite natural, I do not remember seeing it.

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  • $\begingroup$ By standard duality you mean it's the pre-dual of the annihilator of $\mu_0$ which is known to be the quotient of $\ell^1_N / V_0$ (where $V_0$ is the 1-dimensional space generated by $\mu_0$) with the induced quotient norm? $\endgroup$
    – ARG
    Apr 10, 2019 at 17:52
  • $\begingroup$ @ARG Here I was rather referring to the duality in linear programming $\endgroup$
    – R W
    Apr 10, 2019 at 19:48

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