We study a general total variation denoising model with weighted L-1 fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel K, and the approximation term is given by the L-1 norm with respect to a non-singular measure with positively lower-bounded L-infinity density. We provide a detailed analysis of the space of non-local BVBV functions with finite total K-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the K-variation and the associated K-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.
