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.