Density of smooth functions in Musielak-Orlicz spaces

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