On denseness of C0 ∞(Ω) and compactness in Lp(x)(Ω) for 0 < p(x) < 1

The main goal of this paper is to prove the denseness of C0 ∞(Ω) in Lp(x) (Ω) for 0 < p(x) < 1. We construct a family of potential type identity approximations and prove a modular inequality in Lp(x) (Ω) for 0 < p(x) < 1. As an application we prove an analogue of the Kolmogorov–Riesz type compactness theorem in Lp(x) (Ω) for 0 < p(x) < 1. © 2018 Independent University of Moscow.