Here is another way to find a fundamental domain. First identify X with [0,1]. You want to pick a single point in each orbit of the action. Just take the smallest one.

Let be more specific. Consider the set A of points x for which the number of points in the orbit of x is equal to the cardinal of G. Your assumption insures that this set is of full measure. Take some Borel subset B in A of full measure in A, such that G acts on B through Borel transformations.

The fundamental domain D is then defined as the image of B by the map $x \rightarrow min\lbrace\ gx\ |\ g\in G\ \rbrace$. Now the image of a Borel set by a Borel map is always measurable. Restricting again to a Borel subset of full measure, we get a Borel fundamental domain for the action.

Here is another way to find a fundamental domain. First identify $X$ with $[0,1]$. You want to pick a single point in each orbit of the action. Just take the smallest one.

Let be more specific. Consider the set $A$ of points $x$ for which the number of points in the orbit of $x$ is equal to the cardinal of $G$. Your assumption insures that this set is of full measure. Take some Borel subset $B$ in $A$ of full measure in $A$, such that $G$ acts on $B$ through Borel transformations.

The fundamental domain $D$ is then defined as the image of $B$ by the map $x \rightarrow \min\lbrace\ gx\ |\ g\in G\ \rbrace$. Now the image of a Borel set by a Borel map is always measurable. Restricting again to a Borel subset of full measure, we get a Borel fundamental domain for the action.