One of the theorems that I make frequent use of in my uncertainty quantification (UQ) research concerns probabilities measures on **Radon spaces**. Without going into details, the UQ methods that I like to work with will work fine if the spaces where your uncertain parameters / functions / models / other gubbins live are Radon spaces, and might fail to work otherwise; therefore, it’s important to know what a Radon space is, and if it’s a serious restriction. (It’s also very useful to know some pithy examples use in response to questions about Radon spaces in talks and poster presentations!)

So… the definition. Consider a topological space (*X*, *T*) and a probability measure *μ*: ℬ(*T*) → [0, 1] defined on the Borel σ-algebra ℬ(*T*) (i.e. the smallest σ-algebra on *X* that contains all the open sets, i.e. those sets that are listed in the topology *T*). The measure *μ* is said to be **inner regular** if, for every ℬ(*T*)-measurable set *E* ⊆ *X*,

This is often informally read as saying that the measure of an arbitrary measurable set can be approximated from within by compact sets. The space (*X*, *T*) is called a **pseudo-Radon space** (my terminology) if every probability measure *μ* on ℬ(*T*) is inner regular, and if the space is also separable and metrizable then it is called a **Radon space** (more standard in the literature, e.g. the book of Ambrosio, Gigli & Savaré on gradient flows in metric spaces).

So, what spaces are (pseudo-)Radon spaces? It turns out that most of the “nice” spaces that one might want to consider are Radon:

- any compact subset of
*n*-dimensional Euclidean space ℝ^{n} is Radon,
- indeed, Euclidean space itself is Radon,
- as is any Polish space (i.e. a separable and completely metrizable space),
- as indeed is any Suslin space (i.e. a continuous Hausdorff image of a Polish space).

This all seems to suggest that non-Radon spaces must be very weird beasts indeed, perhaps spaces that are topologically very large, so much so that they cannot be the image of a separable space. However, there are, in fact, “small” examples of non-Radon spaces. Since just one counterexample will suffice, it’s enough to find a single example of a non-inner-regular measure on space to show that it is not (pseudo-)Radon.

Continue reading “Radon and non-Radon spaces”