Mass Flow For Noncompact Manifolds

摘要

The group of homeomorphisms acts on the space of measures on a manifold M in such a way that, for each measure μ on M and each homeomorphism h: M → M, the action h_*μ of h on μ is defined by h_*μ(E) = μ(h~(-1)(E)) for each Borel set E is contained in M. The group of homeomorphisms preserving a given measure is just the stabilizer of that measure under the action. In 1941, Oxtoby and Ulam [12] characterized the orbit of standard Lebesgue measure on the unit cube under this action. Since then, Oxtoby and Ulam's result has been of enormous importance for the study of groups of measure-preserving homeomorphisms.