On denseness of $C_0^\infty(\Omega)$ and compactness in $L_{p(x)}(\Omega)$ for $0<p(x)<1$

The main goal of this paper is to prove the denseness of $C_0^\infty(\Omega)$ in $L_{p(x)}(\Omega)$ for $0<p(x)<1$. We construct a family of potential type identity approximations and prove a modular inequality in $L_{p(x)}(\Omega)$ for $0<p(x)<1$. As an application we prove an analogue of the Kolmogorov–Riesz type compactness theorem in $L_{p(x)}(\Omega)$ for $0<p(x)<1$.