I'm working through the book "Ergodic Theory, Independence and Dichotomies by David Kerr and Hanfeng Li." They define a group action on a probability space to be free if there is a G-invariant set $X_0\subseteq X$ with $\mu(X_0)=1$ such that if $sx=x$ for some $x\in X_0$ and $s\in G$ then $s=e$ the identity. It is also claimed that if the action is free then $sA\cap A=\emptyset$ for some measurable nonnull subset $A$.
However this does not seem intuitive to me. Suppose you had a $s\in G$, $s\neq e$, and $q\in sA\cap A$, so $q\in A$ and there is an $a\in A$, $q\neq a$ such that $sa=q$. Then the intersection $sA\cap A$ would contain $q$, yet no point needs to be stabilized by the action.