Erdős-Sierpinski duality in locally compact Polish groups (e.g. $\mathbb{R}^n$)

Question

Erdős-Sierpinski mapping for a locally compact Polish group $G$ is a bijection $f$ from $G$ to $G$ such that $A$ is a null set in $G$ with respect to the Haar measure if and only if $f(A)$ is a meager set in $G$.

Erdős and Sierpinski proved that assuming CH, such a mapping exists for $G=\mathbb{R}$ and it is well-known that it also holds for $G=2^\omega$. However, what about for general locally compact Polish group $G$? (For instance, $\mathbb{R}^n$.)

According to the following paper, for any Polish group, there is no such mapping that also preserves the group addition. This fact was known previously for $G=\mathbb{R}$ and $G=2^\omega$.

Balka, Richárd, Duality between measure and category in uncountable locally compact abelian Polish groups. Real Anal. Exchange36(2010/11/2011), no.2, 245–256.

Answer 1

The answer (under CH) is yes for $\mathbb{R}^n$, and probably yes for any Polish space equipped with a reasonable measure, and doesn't have anything to do with the group structure. The reason is there is both a measure isomorphism and a "category isomorphism" between $\mathbb{R}$ and $\mathbb{R}^n$, which allows you to transfer the result from $\mathbb{R}$ to $\mathbb{R}^n$. A measure isomorphism is a bijection $f:\mathbb{R}\rightarrow\mathbb{R}^n$ s.t. $A\subseteq\mathbb{R}$ is null iff $f(A)$ is null; for a category isomorphism just change null to meager.

To find a measure isomorphism, first find a measure isomorphism between $\mathbb{R}$ and the Cantor space $2^\omega$, and then note that $2^\omega$ and $(2^\omega)^n$ are isomorphic. Of course measure isomorphisms are preserved under product, but I'm not sure about category, so for category isomorphism my thought is to use the following fact:

The Baire space $\omega^\omega$ (homeomorphic to the irrationals $\mathbb{R}\setminus\mathbb{Q}$) is the unique zero-dimensional perfect Polish space whose any compact subset has empty interior.

This implies that if you delete any uncountable dense meager $F_\sigma$ set $D$ from $\mathbb{R}$ (e.g. the union of $\mathbb{Q}$ and the middle-thirds Cantor set), what remains is homeomorphic to the Baire space. If you delete the union $E$ of all rational horizontal or vertical lines from $\mathbb{R}^2$, you have deleted a meager set, and what remains is homeomorphic to the Baire space. Choosing any bijection between $D$ and $E$, we see $\mathbb{R}$ and $\mathbb{R}^2$ are category isomorphic. The same construction works for $\mathbb{R}^n$.

In general, one can use the fact that there is a unique standard probability space, and if $X$ is a perfect Polish space with a base $\mathcal{B}$, then $X\setminus D$ is homeomorphic to the Baire space, where $D=\bigcup\{\overline{U}\setminus U:U\in\mathcal{B}\}$.