In Johnstone's book "Topos theory" there is an interesting example of a topos based on a measure space, see 6.62(ii). If $(X,\Sigma,\mu)$ is a measure space, then the site consists of the poset $\Sigma$, with a coverage generated by countable families of inclusions $(B_i \to B)$ such that $B - \cup B_i$ has measure 0.

The fascinating thing about this topos is that the Dedekind reals in this topos is the sheaf of random variables on $X$, i.e. equivalence classes of measure real-valued functions modulo almost-everywhere equality.

In Johnstone's book "Topos theory" there is an interesting example of a topos based on a measure space, see 6.62(ii). If $(X,\Sigma,\mu)$ is a measure space, then the site consists of the poset $\Sigma$, with a coverage generated by countable families of inclusions $(B_i \to B)$ such that $B - \cup B_i$ has measure 0.

The fascinating thing about this topos is that the Dedekind reals in this topos is the sheaf of random variables on $X$, i.e. equivalence classes of measurable real-valued functions modulo almost-everywhere equality.