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 $$ \|f\|_{\mu,p}\triangleq \inf\left\{ \lambda >0 : \int_{x \in \overline{\Omega}} \left(\frac{|f(x)|}{\lambda}\right)^{p(x)} \mu(dx) \leq 1 \right\} . $$


Yes this is true (in the variable exponent's case which was the objective of the question).

See Theorem 3.4.12 of this book: "Lebesgue and Sobolev spaces with variable exponents" by "Lars Diening, Petteri Harjulehto,Peter Hästö, and Michael Růžička".


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