TITLE: A nice subcategory of the category of measurable spaces
QUESTION [21 upvotes]: Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties?

The real line equipped with the Lebesgue $\sigma$-algebra is nice.
Any translation $\mathbb R \to \mathbb R$ is nice. (Ideally, any continuous map $\mathbb R \to \mathbb R$ is nice, but right now I'm not picky.)
Any finite or countable set equipped with the discrete $\sigma$-algebra is nice.
Nice measurable spaces and nice maps form an elementary topos.

Observations:

If such a topos exists, it is Boolean but does not satisfy the axiom of choice. To prove the latter: Construct the unit circle as a quotient of $\mathbb R$, and then consider the quotient of the unit circle by an irrational rotation. This quotient map cannot split, because if it did the image of its section would be a Vitali set.
If such a topos does not exist, any proof of its nonexistence must rely on the axiom of choice, since in the Solovay model the category of discrete measurable spaces satisfies the conditions.

Edited to add: I would also be interested in the answer to the same question with the Lebesgue $\sigma$-algebra replaced by the Borel $\sigma$-algebra.

REPLY [3 votes]: Take the category of measurable locales, equip it with its natural Grothendieck topology,
and take the topos of sheaves of sets on the resulting site.
(Apply standard disclaimers about universes, coaccessibility, or cosmall sheaves to avoid size issues.)
The resulting Grothendieck topos contains the category of measurable locales
(or, equivalently, the category of measurable spaces,
see https://mathoverflow.net/a/20820 for a detailed description of the latter, and https://mathoverflow.net/a/49542 for more information).
Therefore it satisfies all of your properties, except for the
part about arbitrary continuous maps: the preimage of a measure 0
set under a continuous map R→R is not necessarily of measure 0,
so you cannot hope to get such maps.
But you do get all continuous maps whose preimage preserves measure 0 sets.