Let $X,Y$ be measurable spaces and $\eta : X \to Y$ be a measurable function. Under what conditions on $\eta$ is the composition with $f : Y \to C$ a well defined operation when $f$ is only specified almost everywhere? Does composition with $\eta$ induce a map between $L^p$ spaces?  I will show how one generally would answer these questions , give an algebraic prespective on the problem(and on measure spaces in general), and give a complete solution when the map $\eta$ is multipilication on the $p$-adic integers.