Representation of the total variation as a γ-limit of BMO-type seminorms

We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the Γ-limit, with respect to the L¹loc topology, of a family of $BMO$-type seminorms is given by 14 times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in L¹loc in terms of the boundedness of the $BMO$-type seminorms.