Let $\mathcal{C}$ be a category equipped with a Grothendieck topology $\tau$, i.e., a site.


Under which conditions on $\mathcal{C}$ can one construct a Borel $\sigma$-algebra, $\sigma\_\tau$, for $\tau$? 


If one *can* construct $\sigma\_\tau$, can one then sensibly define a "finitely additive measure" $\mu$ on $\mathcal{C}$ for $\sigma\_\tau$ that takes values in an additive category $\mathcal{A}$? 


Which additional conditions on $\mathcal{A}$ would ensure that such a *categorical measure*, if one is possible, behaves like the usual $\sigma$-additive measures?



Measures on sites
I guess that a good reference might be Olivier Leroy's theory, developing measure theory for locales (a special kind of topoi). Unfortunately Leroy passed away dramatically in 1996 and the paper has been typeset by Claire Voisin and Jean Malgoire, and can be dowloaded from [arXiv](https://arxiv.org/abs/1303.5631). The paper is in French and the title can roughly be translated as "Measure theory in locales or: hidden intersections in Banach-Tarsky's paradox" (the original title is *Théorie de la mesure dans les lieux réguliers ou : Les intersections cachées dans le paradoxe de Banach-Tarski*).


