Extending Fubini Measures

Let $C\subseteq M$ be stably embedded in a structure $\text{cM}=(M;\dots )$. We consider {\em Fubini measures} on the subcategory $\text{Def}(C)$ of the category $\text{Def}(\text{cM})$ of definable sets in $\text{cM}$, 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 $\text{T}$ of transseries with $C=\mathbb{R}$, and to differentially closed fields with constant field $C$.