BMO-type functionals, total variation, and $\Gamma$-convergence

We study the BMO-type functional $\kappa_{\varepsilon}(f,\mathbb R^n)$, which can be used to characterize BV functions $f\in BV(\mathbb R^n)$. The $\Gamma$-limit of this functional, taken with respect to $L^1_{\mathrm{loc}}$-convergence, is known to be $\tfrac 14 |Df|(\mathbb R^n)$. We show that the $\Gamma$-limit with respect to $L^{\infty}_{\mathrm{loc}}$-convergence is $$\tfrac 14 |D^a f|(\mathbb R^n)+\tfrac 14 |D^c f|(\mathbb R^n)+\tfrac 12 |D^j f|(\mathbb R^n),$$ which agrees with the ``pointwise'' limit in the case of SBV functions.