Smooth measures and the canonical retraction in NIP theories

We show that the results proved by Simon on the canonical retraction $F_M$ from the space of $M$-invariant types onto the space of types finitely satisfiable in $M$ remain true over measures. We also make another construction of the canonical retraction for measures, mimicking what Simon did for types, and show that it coincides with Simon’s canonical retraction for measures. To do so, we make extensive use of smooth measures.