In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The only difference is that the continuity condition for the measures is regularity from below on open sets.
From what I know if I started from a hausdorff topological space, got the Borel measurable space, then the locale would at least contain a discrete copy of the starting topological space since the sets containing just a Singleton are $\sigma$-ideals.
I want to know if someone explicited this construction for some measurable space and use it anywhere, more precisely I would like to know if it obvious that such a locale isn't spatial. I have some trouble computing the $\sigma$-ideals of such a space even in basic examples.
