Modeling FO-limits for monadically stable sequences

TL;DR

This paper demonstrates that for monadically stable theories, FO-convergent sequences of structures have modeling limits, and establishes a Borel removal lemma for monadically stable structures, advancing understanding of limits in model theory.

Contribution

It introduces a method to produce modeling limits for FO-convergent sequences in monadically stable classes, a novel result in the area.

Findings

Every FO-convergent sequence in a monadically stable class admits a modeling limit.

A Borel removal lemma is established for monadically stable Lebesgue structures.

The approach leverages probability measures on definable sets within saturated models.

Abstract

We show that given a monadically stable theory $T$, a sufficiently saturated $\mathbf M \models T$, and a coherent system of probability measures on the $\sigma$-algebras generated by parameter-definable sets of $\mathbf M$ in each dimension, we may produce a totally Borel $\mathbf B \prec \mathbf M$ realizing these measures. Our main application is to prove that every FO-convergent sequence of structures (with countable signature) from a monadically stable class admits a modeling limit. As another consequence, we prove a Borel removal lemma for monadically stable Lebesgue relational structures.