Submission date: 27 July 2024

Abstract:

In recent work, Harman and Snowden introduced a notion of measure on a Fraïssé 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(s^2 log{s})).

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2407.19131: pdf, ps.