Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a function of bounded variation. Then do we have \begin{eqnarray} \int\limits_{\mathbb{R}^{2n}}|f(x)-f(y)|\phi_{\epsilon}(|x-y|)dx dy \leq \epsilon TV(f) \end{eqnarray} Where $\phi_{\epsilon}: \mathbb{R} \rightarrow \mathbb{R}$ is Friedrichs mollifier.
If so how to prove it?
P.S.: Definition of total variation in multi dimension can be found here