Extending Fubini Measures

TL;DR

This paper studies Fubini measures in model theory, showing they extend uniquely from a subcategory of definable sets over a stably embedded subset to a larger class, with applications to differential fields and transseries.

Contribution

It establishes the unique extension of Fubini measures from definable sets over a stably embedded subset to fiberable sets, linking to co-analyzability and applications in differential algebra.

Findings

Fubini measures extend uniquely to fiberable sets over C.

Application to differential fields and transseries.

Connection between fiberability and co-analyzability.

Abstract

Let $C\subseteq M$ be stably embedded in a structure $\cM=(M;\dots)$. We consider {\em Fubini measures} on the subcategory $\Def(C)$ of the category $\Def(\cM)$ of definable sets in $\cM$, with ``Fubini" signaling good behaviour in definable families. We show that such a Fubini measure extends uniquely to the larger subcategory of $\Def(\cM)$ whose objects are the sets that are ``fiberable over $C$". In cases of interest ``fiberable over $C$" coincides with ``co-analyzable relative to $C$." This applies in particular to the differential field $\T$ of transseries with $C=\R$, and to differentially closed fields with constant field $C$.