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Ergodic decomposition for measures quasi-invariant under a borel action of an inductively compact group. (English, Russian) Zbl 1302.37004
Sb. Math. 205, No. 2, 192-219 (2014); translation from Mat. Sb. 205, No. 2, 39-70 (2014).
The first main result of the article states that for measurable action $${\mathfrak{T}}$$ of an inductively compact group $$G$$ on a standard Borel space $$X$$ and positive real-valued multiplicative cocycle $${\rho}$$ over $${\mathfrak{T}}$$ with some continuous property there exist a Borel set $${\widetilde{X}\subset X}$$ and a surjective Borel map $${\pi:\widetilde{X}\to\mathfrak{M}_{\mathrm{erg}}(\rho,\mathfrak{T})}$$ such that for any measure $${\nu\in\mathfrak{M}(\rho,\mathfrak{T})}$$ the following ergodic decomposition holds $\nu=\int_{\mathfrak{M}_{\mathrm{erg}}(\rho,\mathfrak{T})}\eta\,d\pi_*\nu(\eta).$ The space $${\mathfrak{M}(\rho,\mathfrak{T})}$$ is a Borel set of probability measures with Radon-Nikodim cocycle $$\rho$$ with respect to $${\mathfrak{T}}$$ and $$\mathfrak{M}_{\mathrm{erg}}(\rho,\mathfrak{T})$$ is a Borel set of ergodic (and indecomposable) probability measures respectively. The proof based on an explicit description of the measurable partition in the sense of Rohlin for a $$\sigma$$-algebra of $$G$$-invariant sets. The second result is the construction of an ergodic decomposition for $$\sigma$$-finite invariant measures. The author proves that the measure class of the decomposing measure on the space of projective measures is uniquely defined.

##### MSC:
 37A15 General groups of measure-preserving transformations and dynamical systems 28D15 General groups of measure-preserving transformations 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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