Given a usual topological space $X$ we define in a natural way the Borel algebra $\mathcal{B}(X)$, i.e., the smallest algebra (or $\sigma$-algebra) of subsets of $X$ containing all open subsets of $X$. This makes $X$ a measurable space, which is very convenient.
Now let $X$ be an object in a category, equipped with a Grothendieck topology. I am only starting to learn this thing, so for me this is basically replacing subsets with inclusion maps. Unions and intersections can be made sense of as pullbacks/pushouts of monic morphisms, and complement of $A$ in $X$ can probably be thought of as a universal subset $B$ such that $A\cup B=X$, if it exists, I don't know.
Question: Is there such a thing as Borel algebra on a Grothendieck topology? Can I give a measure space structure to an object of a category beyond the category of sets/spaces?
I have seen a few posts on MSE and MO which go the other way round, i.e., start from a measure space and topologize it etc. That is not what I am interested in.
I am completely illiterate in abstract algebra and category theory, so, please, be forgiving and use as simple arguments and terminology as possible. A positive-ish answer to this question would be a great motivation for me to study the subject properly.
Thank you.