For a recent result see Alex Simpson's "Measure, Randomness and Sublocales". He shows that in locale theory it is possible to have an isometry-invariant measure on $\mathbb{R}^n$ for which all subsets are measurable. He also defines the locale of random sequences as the sublocale of those sequences which satisfy all measure 1 properties. The locale of random sequences is not empty (but has no points!), and in fact its measure is 1. All of this is quite impossible if you insist that spaces must have lots of points.