# Reference Request: $L^p(x)$/(Musielak–Orlicz space) analogue of classical $L^p$ result

Fix a non-empty open domain $$\Omega\subseteq \mathbb{R}^d$$ with compact closure, and a finite Borel measure $$\mu$$ on its closure $$\overline{\Omega}$$.

In Halmos' book it is shown that:

Classical Result: *For any bounded function $$f\in L^p_{\mu}(\Omega;\mathbb{R})$$ and every $$\epsilon >0$$, there exists a continuous function $$g$$ such that $$\int_{x \in \Omega} |f(x)-g(x)|^p \mu(dx) < \epsilon.$$*

Reference/Question: Is there an analogue of this result for the Musielak–Orlicz spaces?

More specifically, I'm wondering if $$p:\Omega\rightarrow \mathbb{R}$$ is a measurable function satisfying the usual conditions (for example see this paper), and $$f \in L^{p(x)}_{\mu}(\Omega),$$ then for every $$\epsilon>0$$, can we find a continuous function $$g$$ on $$\overline{\Omega}$$ such that $$\|f-g\|_{\mu,p}<\epsilon,$$

Background: Where in the above: $$L^{p(x)}_{\mu}(\Omega)\triangleq \left\{ f:\Omega \rightarrow \mathbb{R}: \mbox{f is measurable and } \int_{x \in \overline{\Omega}} |f(x)|^{p(x)} \mu(dx)<\infty \right\},$$ and can be seen to be a Musielak–Orlicz space space under the Luxemburg norm $$\|\cdot\|_{\mu,p}$$ defined by $$\|f\|_{\mu,p}\triangleq \inf\left\{ \lambda >0 : \int_{x \in \overline{\Omega}} \left(\frac{|f(x)|}{\lambda}\right)^{p(x)} \mu(dx) \leq 1 \right\} .$$