Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in this paper, where I also established a similar result for the volume-preserving PL pseudogroup. In the PL case, the corresponding decoration on the manifold is a piecewise constant volume form.

In my opinion, the most exciting result on this theme is the Ulam-Oxtoby theorem. (But Moser's version is the most useful one and the most elegant.) The theorem is that if you have a topological manifold with any Borel measure that has no atoms and no bald spots, then it is modeled on the pseudogroup of volume-preseserving homeomorphisms. For example, you can start with Lebesgue measure in the plane and add uniform measure on a circle, and there is a homeomorphism that takes that measure to Lebesgue measure.

For a long time I have wondered about the pseudogroup of volume-preserving Lipschitz maps. The question is whether there is a corresponding cone of measures, and if so, how to characterize it.

Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in my paper A volume-preserving counterexample to the Seifert conjecture (Comment. Math. Helv. 71 (1996), no. 1, 70-97), where I also established a similar result for the volume-preserving PL pseudogroup. In the PL case, the corresponding decoration on the manifold is a piecewise constant volume form.

In my opinion, the most exciting result on this theme is the Ulam-Oxtoby theorem. (But Moser's version is the most useful one and the most elegant.) The theorem is that if you have a topological manifold with any Borel measure that has no atoms and no bald spots, then it is modeled on the pseudogroup of volume-preseserving homeomorphisms. For example, you can start with Lebesgue measure in the plane and add uniform measure on a circle, and there is a homeomorphism that takes that measure to Lebesgue measure.

For a long time I have wondered about the pseudogroup of volume-preserving Lipschitz maps. The question is whether there is a corresponding cone of measures, and if so, how to characterize it.

Yes! This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in this paper, where I also established a similar result for the volume-preserving PL pseudogroup.

Yes for "hope" 1. This theorem was proven by Moser using volume-preserving flows. A manifold with a volume form is the same thing as a manifold with an atlas of charts modeled on the volume-preserving diffeomorphism pseudogroup. He found an argument that can be adapted to either the symplectic case or the volume case. I cited this result in this paper, where I also established a similar result for the volume-preserving PL pseudogroup. In the PL case, the corresponding decoration on the manifold is a piecewise constant volume form.

In my opinion, the most exciting result on this theme is the Ulam-Oxtoby theorem. (But Moser's version is the most useful one and the most elegant.) The theorem is that if you have a topological manifold with any Borel measure that has no atoms and no bald spots, then it is modeled on the pseudogroup of volume-preseserving homeomorphisms. For example, you can start with Lebesgue measure in the plane and add uniform measure on a circle, and there is a homeomorphism that takes that measure to Lebesgue measure.

For a long time I have wondered about the pseudogroup of volume-preserving Lipschitz maps. The question is whether there is a corresponding cone of measures, and if so, how to characterize it.