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Pushforward of invariant measures (equivariant 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 $ … (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. …
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