  We consider bipartite graphs definable in o-minimal structures, in which the
edge relation $G$ is a finite union of graphs of certain measure-preserving
maps.
  We establish a fact on the existence of definable matchings with few short
augmenting paths. Under the additional assumptions that $G\subseteq [0,1]^n$
and 2-regularity, this yields the existence of definable matchings covering all
vertices outside of a set of arbitrarily small positive measure (Lebesgue
measure of the standard part). As an application we obtain an approximate
2-cancellation result for the semigroup of definable subsets of $[0,1]^n$
modulo an equivalence relation induced by measure-preserving maps.
