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I recently have come across the total variation metric on probability measures as a dual to a certain space of continuous functions. My question is, what is the use of this metric and what does it represent?

As a comparative example, the Levy-Prokhorov metric topologies the weak$^{\star}$ topology on the boundary of the unit ball in the space of finite signed measures and represents convergence in distribution. But what does total variation represent statistically/probabilistically?

AB_IM
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One of the several theorems known as the Riesz Representation Theorem tells us that if $X$ is a locally compact Hausdorff topological space, then any continuous functional $\phi$ on the space $C_c(X)$ of compactly supported continuous functions on $X$ can be represented uniquely as a bounded Radon measure $\mu_\phi$ such that $\phi(f) = \int f \mathrm{d} \mu_\phi$ for all $f \in C_c(X)$. This isomorphism will have some cool properties.

Firstly, the total variation of $\mu_\phi$ is equal to the norm of $\phi$. The positive functionals are equivalent to the nonnegative measures (as opposed to the “signed measures”). Finally, the positive linear functionals with $\| \phi \| = 1$ are equivalent to the space of Radon probability measures on $X$.

Finally, note that if $X$ is compact, then $C_c(X) = C(X)$.

AJY
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  • Yes, this I know (asI implicitly mention in the question). However, from the statistical/probabilistic perspective I see no motivational explanation here. – AB_IM Apr 29 '20 at 09:52
  • @ArtificialHuman I’m no statistician, but perhaps it could describe how certain random variables behave differently under different measures. Perhaps if you told us where you’d seen this referenced it could be of use. – AJY Apr 29 '20 at 09:56
  • Oh I came across it exactly in this way, through the dualization of $C_c(X)$ with the inductive limit topology. – AB_IM Apr 29 '20 at 10:12
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    @ArtificialHuman Then I suppose it could be useful for understanding how a measure integrates, since if $X, Y$ are two $L^\infty$ random variables on a probability space $(\Omega, \mu)$, then $\left| \int X \mathrm{d} \mu - \int Y \mathrm{d} \mu \right| \leq | X - Y |_{\infty} \cdot | \mu |$. – AJY Apr 29 '20 at 10:20
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    Oh this is interesting. Do you have a reference to this (duality-type bound). I've never seen it before (except for the Hölder analogue) – AB_IM Apr 29 '20 at 10:31
  • @ArtificialHuman it’s just the inequality $\left| \int f \mathrm{d} \mu \right| \leq \int |f| \mathrm{d} \mu$. – AJY Apr 29 '20 at 20:40