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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.

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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