Let $(X,d,m)$ and $(Y,\rho,\mu)$ be doubling metric measure spaces. The seminal work of Korevaar and Schoen discussed generalizations of $L^p$ spaces for maps from $X$ to $Y$. Standard results from Analysis show that when $X=\mathbb{R}^d$ and $Y=\mathbb{R}^D$ then $C_c(\mathbb{R}^d,\mathbb{R}^D)$ (continuous functions with compact support) is dense in the Bochner-Lebesgue space $L^1(\mathbb{R}^d,\mathbb{R}^D)$.

I'm wondering, if this is true for the $L^p(X,Y)$ spaces of of Korevaar and Schoen and more general if there are extensions of $L^{p(x)}$ spaces, Musielak-Orlicz spaces, between $(X,d,m)$ and $(Y,\rho,\mu)$ for which $C(X,Y)$ is dense in those spaces.


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