Conull subspace containing orbit of an (ergodically acting) group

Question

I should probably start with a warning that this is my first post in this board and that I am sorry, if it is not up to standards. It would be great, if you could let me know how to improve the post.

I am currently working towards my master thesis, which is mainly concerned with the following paper: Chatterji, Iozzi, Fernós - The median class and superrigidity of actions on CAT(0) cube complexes. More precisely, I try to understand the construction of the (measurable, $\Gamma$-equivariant) boundary map $$ \varphi\colon B \to \partial X, $$ where $B$ is a strong $\Gamma$-boundary for a group $\Gamma$ and $\partial X$ is the Roller boundary of a finite dimensional, locally countable CAT(0) cube complex $X$. The group action on $X$ is assumed to be essential and non-elementary. However, I only mention this for completeness sake. I think the precise setting should not enter in my actual question, because as far as I can see it is completely measure theoretic.

During the construction, there is a particular recurring argument I just cannot get my head around. I think, it should follow from general measure theoretic arguments, but my background in that field is not strong enough to pin it down. If I would have to formulate it as a Lemma, it would take the following form:

Let $(X,\mu)$ be a probability space on which a group $\Gamma$ acts ergodically. Let $B \subset X$ be measurable and $\mu(X \setminus B)=0$. Then $B$ contains a $\Gamma$ orbit.

I would be interested in any hint pointing towards a proof of this statement or towards any counterexample. I would also be interested in any restrictions necessary to make the above statement work. In particular, I was unable so far to collect all the assumptions I have on the group $\Gamma$. However, I am certain that a locally compact and a second countable topology are necessary and can be used, if necessary. Additionally, $(X,\mu)$ can be assumed to be a Lebesgue space.

Your help is very much appreciated. Thank you very much for your time and effort.

Best regards,

Tim

Answer 1

Formally, your question was answered in the comments, and I will not repeat the answer here, as it simply indicates that your initial question was "wrong". Possibly you just missed some "up to null set" comment (or hidden assumption) somewhere in the text. Nevertheless, if you are genuinely interested in delicate and fundamental questions of the type indicated in the OP, you may find some answers reading

Feldman, Jacob; Hahn, Peter; Moore, Calvin C. Orbit structure and countable sections for actions of continuous groups. Adv. in Math. 28 (1978), no. 3, 186–230.

Edit: In respond to a comment by YCor, I explain how applying the results of the above mentioned paper answers the OPed question.

A main result of the FHM paper is Theorem 2.8 which states that for every lcsc group $\Gamma$, for every analytic action of $\Gamma$ on a space $X$, and for every invariant measure class $\mu$ on $X$, there exists a measurable subset $N$ in $X$ and an identity neighborhood $U$ in $\Gamma$ such that $\Gamma N$ is conull in $X$ and for every $x\in N$, $Ux\cap N=\{x\}$. Replacing $X$ with $\Gamma N$ we may assume that $X=\Gamma N$. It is now an easy exercise to see that, unless the $\Gamma$ action on $X$ has discrete orbits, $N$ must be a null set. In this case, setting $B=X-N$ we get a contradiction to the boldfaced statement in the OP.

I didn't elaborate on this before, as Mikael de la Salle already gave an explicit counter-example in a comment: the action of $\mathbb{R}/\mathbb{Z}$ on itself (and $N=\{1\}$). Another, slightly less transparent, example to keep in mind is the following: Consider an irrational flow of $\mathbb{R}$ on the two torus $\mathbb{R}/\mathbb{Z}\times \mathbb{R}/\mathbb{Z}$ and let $N=\mathbb{R}/\mathbb{Z}\times \{0\}$.

But, let me emphasize: I do not think these type of questions (important us they are) are really relevant for the understanding of the Chatterji-Iozzi-Fernós paper.