Pseudoquotient extensions of measure spaces

Abstract: 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 $(X,\Sigma,\mu)$ 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 $\mathcal P (X,S)$.