Pseudoquotient extensions of measure spaces

A space of pseudoquotients $\mathcal{P}(X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f)\sim (y,g)$ if $gx=fy$. In this note we assume that $(X,\mathrm{\Sigma },\mu )$ is a measure space and that $S$ is a commutative semigroup of measurable injections acting on $X$ and investigate under what conditions there is an extension of $\mu$ to $\mathcal{P}(X,S)$.