This is about exercise 3.3 from Ambrosio, Fusco and Pallara Functions of bounded variation and free discontinuity problems. I struggled with this for some time.
Let $\Omega$ be an open set and $u \in L^1_{loc}(\Omega)$. Show that the totalvariation $V(u, \cdot)$ is the least Borel measure $\mu$ s.t.
$$ \int_K \frac{|u(x+y) - u(x)|}{|y|}\mathrm{d}x \leq \mu\big(\{x \in \Omega: \text{dist}(x,K) < |y|\}\big). $$
for any compact set $K \subset \Omega$ and any $y \in \mathbb{R}^N\setminus \{0\}$ such that $|y| < \text{dist}(K, \partial \Omega)$.
Here the total variation is defined on open sets $A$ like usual by
$$ V(u,A) := \sup\left\{ \int_A u(x)\text{div}\phi(x) \mathrm{d}x: \phi \in C^1_c(A), \left\lVert\phi\right\rVert_\infty \leq 1\right\}.$$
Does anyone have an idea? Particular the part about that the total variation is the least Borel measure that has this property has me kind of stumbled.