The recurrence theorem of Halmos is well known in the case of a non-singular endomorphism $T$ of a measured space $(X,\mathcal B,\mu)$. A measurable subset $A$ is contained in the conservative part (mod $\mu$) if and only if $$ \sum_{n \geq 0} 1_B \circ T^n = \infty $$ holds a.e. in $B$ (where $1_B$ stands for the characteristic function) for any measurable $B \subset A$.

See e.g. Aaronson's book. Now I have never seen a proof of this theorem in the case of a non-singular group operation (where the group is locally compact and second countable). In fact I have never seen it stated in this form but I believe it must be true.

Does someone know of a reference?

A related theorem is Hopf's theorem which states that the conservative part is (mod $\mu$) the set of all $x$ such that $$ \int f(hx) d\eta(h) = \infty $$ where $f \in L^1(\mu)$ is $>0$ and $\eta$ is some Haar measure on $H$. I would be equally happy with a proof of Hopf's theorem instead.

Thank you.