I am primarily thinking of $\mathsf{Ran}(\mu)$ as the sublocale given by the quotient $U \equiv V$ whenever $\mu(U) = \mu(U \wedge V) = \mu(V)$, and secondarily as the smallest sublocale of the same measure, though Simpson's characterization with Thm 2 does require that the locale be fitted. I doubt that the theorem generalizes to infinite measures (if I have discussed infinite measures then I didn't notice it and it is an accident).

Oops, I meant $\mu(\cdot > q) = \frac{1}{2+q}$.

Thanks for your helpful remarks! I intended that for any locale $A$ and opens $U, V$ of $A$, $U < V$ means $U \le V$ and $\neg (V \le U)$. Perhaps there is a negation-less way to phrase everything. I think the probability distribution $\mu$ on $R$ given by $\mu(\cdot > q) = 1 - \frac{1}{2 + q}$ is such that $R$ is random w.r.t. $\mu$.

@IngoBlechschmidt: Yes, I think that example works; thanks! I was previously confused as to whether the points of $\mathbb{R}$ formed a set, and thus whether the example you have is indeed inductively generated, since it has $\mathbb{R}$-many axioms, and also joins over all of $\mathbb{R}$. But I now believe that these can be made predicatively acceptable because points of $\mathbb{R}$ "essentially" form a set.

@SimonHenry: Thanks so much for referring me to that paper! I was not previously aware of it, and basically answers my question. There was also an error in my question: though the points of $\mathbb{N}^\mathbb{B}$ do not form a set, they "essentially" do (Curi calls this "smallness"), and hence the covers in my example at the end could be inductively generated, hence giving a non-spatial inductively generated formal space.