We introduce a class of concentrated p-Lévy integrable functions approximating the
unity, which serves as the core tool to characterize the Sobolev spaces and the space of functions
of bounded variation in the spirit of Bourgain-Brezis-Mironescu. We provide this characterization
for a class of unbounded domains satisfying the extension property. This characterization will play
a decisive role while studying the asymptotic of solutions to elliptic IntegroDifferential Equations (IDEs).
We will focus on (non)local elliptic problems subject to Neumann type condition.