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Bi-invariant functions on the group of transformations leaving a measure quasi-invariant. (English. Russian original) Zbl 1309.22017
Sb. Math. 205, No. 9, 1357-1372 (2014); translation from Mat. Sb. 205, No. 9, 145-160 (2014).
This paper studies various properties of the group \(Q\) of transformations of a Lebesgue space for which the measure is quasi-invariant, and the subgroup \(P\) of transformations preserving the measure. The group \(Q\) has been given a topology by several authors in different ways, including Yu. A. Neretin [Russ. Acad. Sci., Sb., Math. 75, No. 1, 197–219 (1993); translation from Mat. Sb. 183, No. 2, 52–76 (1992; Zbl 0774.58006)], A. S. Kechris [Classical descriptive set theory. Berlin: Springer-Verlag (1995; Zbl 0819.04002)] and V. Pestov [Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1123.37003)], and one of the results here shows that these topologies coincide. A canonical form for the double cosets \(P\backslash Q/P\) is found, and the space of \(P\)-biinvariant continuous functions on \(Q\) is identified with the space of functionals of the distribution of a Radon–Nikodym derivative.
MSC:
22E66 Analysis on and representations of infinite-dimensional Lie groups
28D99 Measure-theoretic ergodic theory
22F10 Measurable group actions
28E99 Miscellaneous topics in measure theory
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