Density of smooth functions in Musielak–Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $$C^\infty _c(\Omega )$$ to be dense in Musielak–Orlicz spaces $$L^\Phi (\Omega where $$\Omega $$ is an open subset $${\mathbb {R}}^d$$ . In particular, we prove that if $$\Phi satisfies condition $$\Delta _2$$ , closure )\cap L^\Phi equal only measure singular points zero. This extends earlier density theorems proved under assumption local integrability which implies As a corollary obtain analogous results generated by double phase functional recover well-known result variable exponent Lebesgue spaces.