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 ℝn 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.
A neat example of a “small” non-pseudo-Radon space is given by equipping the real line ℝ (or any subset of it) with a slightly different topology, namely the lower limit topology (LLT). The LLT on ℝ is the topology of “convergence from the right”; it is the topology generated by the basis of open sets (intervals) of the form [a, b), where a and b are real numbers. (Contrast this with the usual Euclidean topology on ℝ, which is the topology generated by the basis of open sets (intervals) of the form (a, b), where a and b are reals.) Now we have two important facts:
- The usual Euclidean topology and the LLT generate the same σ-algebra on ℝ. This can be seen by noting that the countable intersection of Euclidean-open sets
is the LLT-open set [a, b), and the countable union of LLT-open sets
is the Euclidean-open set (a, b). Hence, when one “closes” either collection of basic open sets to form a σ-algebra, one gets a collection of sets that must include the other collection of basic open sets; therefore, the two generated σ-algebras are the same. This means that a measure such as standard Gaussian measure (the normal distribution) is a well-defined Borel probability measure on (ℝ, LLT). Similarly, the uniform distribution (Lebesgue measure) on the unit interval [0, 1], restricted to the Borel sets, is a well-defined Borel probability measure on ([0, 1], LLT).
- Every compact subset of (ℝ, LLT) must be a countable set. To see this, consider a non-empty compact subset K of (ℝ, LLT). For each x ∈ K, consider the following open cover of K:
Since K is compact, this cover has a finite subcover, and hence there exists a real number a(x) such that the interval (a(x), x] contains no point of K other than x. Now choose a rational number q(x) ∈ (a(x), x]. Since the intervals (a(x), x], parametrized by x ∈ K, are pairwise disjoint, the function q: K → ℚ is injective, and so K is a countable set.
Put together, these facts mean that standard Gaussian measure on (ℝ, LLT) is not inner regular (the measure of ℝ is 1 yet the measure of every compact subset of ℝ is 0), and so (ℝ, LLT) is not a pseudo-Radon space; uniform measure on ([0, 1], LLT) faile to be inner regular for the same reason. As it happens, (ℝ, LLT) is separable but not metrizable, so it fails two out of the three criteria to be a Radon space as Ambrosio, Gigli & Savaré use the term.