"On yet another analog of Riesz brothers theorem for Sobolev space"

Abstract: We show that the quotient space BV/W^{1,1}
is isomorphic to the space of bounded borel measures. Here BV denotes
the space of functions of bounded variation and W^{1,1}
the Sobolev space of functions with integrable gradient
on regular domain. One can see this as an analog of
Pełczyński's result that dual to the space of C^1 -smooth
functions is a separable perturbation of the space of measures.
Main ingredients of a proof are G. Alberti rank one theorem
and extension/averaging results for Sobolev and BV spaces.