Non-absolutely convergent integrals with respect to distributions

We define an integral of a function with respect to a distribution. In case that the underlying distribution is just the Lebesgue measure, the definition leads to a new non-absolutely convergent integral which is wider than the Denjoy–Perron integral. We present a version of the Gauss–Green theorem where the new integral is used for both interior and boundary terms. As a by-product, we characterize the predual Sobolev space \(W^{-1,1}\) .