Exercise 12.17: Total Variation Norm

chapter 12

In Exercise 3.13, we established that \begin{equation} \TV(\P, \Q) = \sup_{f \colon \X \to [0, 1]} \int f(p - q) \isd \mu. \end{equation} Therefore, \begin{equation} 2\TV(\P, \Q) = \sup_{f \colon \X \to [-1, 1]} \int f(p - q) \isd \mu = \sup_{\norm{f}_\infty \le 1} \int f(p - q) \isd \mu. \end{equation}