Upper bounds for measures on distal classes

In recent work, Harman and Snowden introduced a notion of measure on a Fra\"iss\'e class F, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and so far only a handful of cases have been worked out. In this paper, we obtain some of the first general results on measures. Our main theorem states that if F is distal (in the sense of Simon), and there are some bounds on automorphism groups, then F admits only finitely many measures; moreover, we give an effective upper bound on their number. For example, if F is the class of ``s-dimensional permutations'' (finite sets equipped with s total orders), we show that the number of measures is bounded above by approximately exp(exp(s² logs)).