Give example of applications in maths domains or more generally of the use of Borel measure that is not a Radon measure

Question

I'm looking for examples of research domains or themes ,where Borel measure that are not Radon measures are used in some ways.

Answer 1

This is a trivial example, but consider the counting measure defined over the Borel sigma algebra. As it is a measure defined over the Borel sigma algebra, it is a Borel measure. It is not a Radon measure, as it is not finite on all compact sets.

Evidently, it has many applications. For example in statistics, where it is used to construct empirical measures.

Answer 2

Geometric measure theory, and all fields where you have to work with Hausdorff measures. Indeed, let $s \in [0,\infty)$ and $\mathcal{H}^s$ denote the $s-$dimensional Hausdorff measure on $\mathbb{R}^n$. Then $\mathcal{H}^s$ is a Borel regular outer measure on $\mathbb{R}^n$, but it is not a Radon outer measure unless $s\ge n$.

Note that, as an example, with Borel regularity of a measure you can apply lots of theory: for example the approximation of outer measures by open and closed sets. The Radon measure allows you to approximate outer measures also with compact sets, instead of closed sets.