We prove that the family of functionals $(I_\delta)$ defined by \[ I_\delta(g) = \mathop{\mathop{\int\int}_{\mathbb{R}^N \times \mathbb{R}^N}}_{|g(x) - g(y)| > \delta} \frac{\delta^p}{|x-y|^{N+p}} dx dy, \quad \forall g \in L^p(\mathbb{R}^N), \] for $p \ge 1$ and $\delta >0$ , $\Gamma$ -converges in $L^p(\mathbb{R}^N)$ , as $\delta$ goes to zero, when $p \ge 1$ . Hereafter $| \; |$ denotes the Euclidean norm of $\mathbb{R}^N$ . We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.