  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 $L^\infty$ density. We provide a detailed analysis of the space
of non-local $BV$ 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.
