  We introduce a large class of concentrated $p$-L\'{e}vy integrable functions
approximating the unity, which serves as the core tool from which we provide a
nonlocal characterization of Sobolev spaces and the space of functions of
bounded variation via nonlocal energies forms. It turns out that this nonlocal
characterization is a necessary and sufficient criterion to define Sobolev
spaces on domains satisfying the extension property. We also examine the
general case where the extension property does not necessarily hold. In the
latter case, we establish weak convergence of the nonlocal Radon measures
involved to the local Radon measures induced by the distributional gradient.
