Is there a natural way to interpret the category of measure spaces as an internal category in the category of topological spaces? any references would be helpful.



Topologizing the category of measure spaces
As demonstrated by Andre Kornell in <http://arxiv.org/abs/1202.2994>,
the category of measurable spaces is closed with respect
to the monoidal structure given by the spatial tensor product (Theorem 9.5).
Points (atoms) of the resulting internal hom
are precisely morphisms of measurable spaces (Corollary 9.12, ibid.).
In particular, we obtain an enrichment of the category
of measurable spaces over itself.
There is a fully faithful forgetful functor from the category
of measurable spaces (= measurable locales) to the category of locales (e.g., see the nLab article [measurable locale](http://ncatlab.org/nlab/show/measurable+locale)),
which gives you the desired enrichment
over (not necessarily spatial) topological spaces.


