Let $d,D$ be positive integers, $p \in [1,\infty)$, $U$ an (non-empty) open subset of $\mathbb{R}^D$, and suppose that $f$ is:
- In the Bochner-Lebesgue space $L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)$, where $\mu$ is a Radon measure on $\mathbb{R}^D$ and $\mathcal{B}(\mathbb{R}^d)$ is the Borel $\sigma$-algebra on $\mathbb{R}^d$
- The $\mu$-essential support of $f$ is in $U$.
Let $\phi$ be an auto-diffeomorphism of $U$. Is it true that $$ f \in L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D) \Leftrightarrow \phi\circ f L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\mathbb{R}^D)? $$ Thoughts/Comments:
- The answer can be worked out in the case of pre-composition by a change of variables formula, but what about this case?
- I know if $\phi$ is linear and $U$ is the entire space, then the answer is yes by one of the basic properties of the Bochner integral (see here for example). However, it's not clear to me in general.