There is a well-known theorem that between any two absolutely continuous Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ there is an increasing triangular transformation $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $ν = T_∗μ$ (see, for example, “Triangular transformations of measures” by Bogachev, Kolesnikov, and Medvedev).

There is another version of this theorem for smooth manifolds (non-compact Moser theorem). If $M$ is a noncompact connected oriented manifold and if $\mu$ and $\nu$ are volume forms on M with $\int_M \mu = \int_M \nu \le \infty$ and if each end of the manifold $M$ has finite $\mu$ volume if it has finite $\nu$ volume and infinite $\mu$ volume if it has infinite $\nu$ volume, then there is a diffeomorphism $\phi: M \to M$, such that $\phi^* \mu = \nu$. (see "Diffeomorphisms and volume-preserving embeddings for noncompact manifolds" by Greene and Shiohama).

I am curious about the equivariant analog of either of these results.

More precisely, assume that a (compact?) group $G$ acts on $\mathbb{R}^n$ and $\mu$ and $\nu$ are two $G$-invariant (smooth) densities. Does there always exist a $G$-equivariant diffeomorphism $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $ν = T_∗μ$?

Could you point out if this result is proved somewhere in the literature? Does the group $G$ has to be compact for it to hold?

Thank you!

There is a well-known theorem that between any two absolutely continuous Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ there is an increasing triangular transformation $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $\nu = T_∗\mu$ (see, for example, “Triangular transformations of measures” by Bogachev, Kolesnikov, and Medvedev).

There is another version of this theorem for smooth manifolds (non-compact Moser theorem). If $M$ is a noncompact connected oriented manifold and if $\mu$ and $\nu$ are volume forms on M with $\int_M \mu = \int_M \nu \le \infty$ and if each end of the manifold $M$ has finite $\mu$ volume if it has finite $\nu$ volume and infinite $\mu$ volume if it has infinite $\nu$ volume, then there is a diffeomorphism $\phi: M \to M$, such that $\phi^* \mu = \nu$. (See "Diffeomorphisms and volume-preserving embeddings for noncompact manifolds" by Greene and Shiohama.)

I am curious about the equivariant analog of either of these results.

More precisely, assume that a (compact?) group $G$ acts on $\mathbb{R}^n$ and $\mu$ and $\nu$ are two $G$-invariant (smooth) densities. Does there always exist a $G$-equivariant diffeomorphism $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $\nu = T_∗\mu$?

Could you point out if this result is proved somewhere in the literature? Does the group $G$ have to be compact for it to hold?