Abstract

Let $C\subseteq M$ be stably embedded in a structure $M=(M;\dots <!--FUNCTION\; APPLICATION-->)$. We consider Fubini measures on the subcategory $\mathrm{Def}<!--FUNCTION\; APPLICATION-->(C)$ of the category $\mathrm{Def}<!--FUNCTION\; APPLICATION-->(M)$ of definable sets in $M$, with “Fubini” signaling good behavior in definable families. We show that such a Fubini measure extends uniquely to the larger subcategory of $\mathrm{Def}<!--FUNCTION\; APPLICATION-->(M)$ 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 $\mathbb{𝕋}$ of transseries with $C=ℝ$, and to differentially closed fields with constant field $C$.

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