Homeomorphism Lifting To Homeomorphism of Lp spaces

Question

Let $(X,d,x_0)$ be a pointed metric space, $\nu$ be a $\sigma$-finite Borel measure on $X$, and $f:X\rightarrow \mathbb{R}^d$ be a homeomorphism satisfying $f(x_0)=0$. Define the space $ L^p(\mathbb{R}^D;X), $ to be the set of measurable functions from $\mathbb{R}^D$ to $X$ satisfying $$ \int_{x \in \mathbb{R}^D} d(f(x),x_0) d\mu(x)<\infty. $$ Does the following map define a homeomorphism: $$ \begin{aligned} \Psi: &L^p(\mathbb{R}^D;X) \mapsto L^p(\mathbb{R}^D;\mathbb{R}^d)\\ g &\mapsto f\circ g \end{aligned} $$ where the right-hand side is the Bochner $L^p$-space?