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?
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 $C_c(X) = C(X)$.