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Regarding what you wrote I would be equally happy with a proof of Hopf's theorem instead, it seems that the very fact that there is "conservative part" for a nonsingular action is unavailable for general lcsc groups. However, there is a "version" of the Hopf Decomposition for the purpose of checking the divergence $$(*)\quad \intop_{G}\frac{dT_{g}^{*}\mu}{d\mu}f\left(T_{g}x\right)dm\left(g\right)=\infty.$$$$(*)\quad \intop_{G}\frac{d\mu\circ g^{-1}}{d\mu}f\left(gx\right)dm\left(g\right)=\infty.$$

This can be done as follows. Pick a lattice in $G$ (if there exists any!) and take the Hopf Decomposition $D\cup C$ w.r.t. this lattice (regarding this lattice as an acting group on its own right, and using the Hopf Decomposition for the countable case as in [1, Proposition 1.6.2]). It is then true that $(*)$ holds for $x\in D$ and, moreover, the set $D$ is independent on the lattice you chose up to null set. See [1, Theorem 1.6.4] and [2, Corollary 2.2].

References

[1] Aaronson, Jon, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997). ZBL0882.28013.

[2] Roy, Parthanil, Nonsingular group actions and stationary S(\alpha )S random fields, Proc. Am. Math. Soc. 138, No. 6, 2195-2202 (2010). ZBL1196.60093.

Regarding what you wrote I would be equally happy with a proof of Hopf's theorem instead, it seems that the very fact that there is "conservative part" for a nonsingular action is unavailable for general lcsc groups. However, there is a "version" of the Hopf Decomposition for the purpose of checking the divergence $$(*)\quad \intop_{G}\frac{dT_{g}^{*}\mu}{d\mu}f\left(T_{g}x\right)dm\left(g\right)=\infty.$$

This can be done as follows. Pick a lattice in $G$ (if there exists any!) and take the Hopf Decomposition $D\cup C$ w.r.t. this lattice (regarding this lattice as an acting group on its own right, and using the Hopf Decomposition for the countable case [1, Proposition 1.6.2]). It is then true that $(*)$ holds for $x\in D$ and, moreover, the set $D$ is independent on the lattice you chose up to null set. See [1, Theorem 1.6.4] and [2, Corollary 2.2].

References

[1] Aaronson, Jon, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997). ZBL0882.28013.

[2] Roy, Parthanil, Nonsingular group actions and stationary S(\alpha )S random fields, Proc. Am. Math. Soc. 138, No. 6, 2195-2202 (2010). ZBL1196.60093.

Regarding what you wrote I would be equally happy with a proof of Hopf's theorem instead, it seems that the very fact that there is "conservative part" for a nonsingular action is unavailable for general lcsc groups. However, there is a "version" of the Hopf Decomposition for the purpose of checking the divergence $$(*)\quad \intop_{G}\frac{d\mu\circ g^{-1}}{d\mu}f\left(gx\right)dm\left(g\right)=\infty.$$

This can be done as follows. Pick a lattice in $G$ (if there exists any!) and take the Hopf Decomposition $D\cup C$ w.r.t. this lattice (regarding this lattice as an acting group on its own right, and using the Hopf Decomposition for the countable case as in [1, Proposition 1.6.2]). It is then true that $(*)$ holds for $x\in D$ and, moreover, the set $D$ is independent on the lattice up to null set. See [1, Theorem 1.6.4] and [2, Corollary 2.2].

References

[1] Aaronson, Jon, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997). ZBL0882.28013.

[2] Roy, Parthanil, Nonsingular group actions and stationary S(\alpha )S random fields, Proc. Am. Math. Soc. 138, No. 6, 2195-2202 (2010). ZBL1196.60093.

Source Link
HUO
HUO
  • 426
  • 3
  • 10

Regarding what you wrote I would be equally happy with a proof of Hopf's theorem instead, it seems that the very fact that there is "conservative part" for a nonsingular action is unavailable for general lcsc groups. However, there is a "version" of the Hopf Decomposition for the purpose of checking the divergence $$(*)\quad \intop_{G}\frac{dT_{g}^{*}\mu}{d\mu}f\left(T_{g}x\right)dm\left(g\right)=\infty.$$

This can be done as follows. Pick a lattice in $G$ (if there exists any!) and take the Hopf Decomposition $D\cup C$ w.r.t. this lattice (regarding this lattice as an acting group on its own right, and using the Hopf Decomposition for the countable case [1, Proposition 1.6.2]). It is then true that $(*)$ holds for $x\in D$ and, moreover, the set $D$ is independent on the lattice you chose up to null set. See [1, Theorem 1.6.4] and [2, Corollary 2.2].

References

[1] Aaronson, Jon, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs. 50. Providence, RI: American Mathematical Society (AMS). xii, 284 p. (1997). ZBL0882.28013.

[2] Roy, Parthanil, Nonsingular group actions and stationary S(\alpha )S random fields, Proc. Am. Math. Soc. 138, No. 6, 2195-2202 (2010). ZBL1196.60093.