Why regularity is important for a Borel measure?

Question

Why the property of being regular for a measure is important and what are some important and essential theorems that use this properties to hold true ?

Thanks in advance.

Answer 1

The main theorem about Borel measures is the Riesz--Markov--Kakutani representation theorem. There are several versions of it, so I am not going to be too precise here. Actually all this answer is mostly discursive, to be taken with a grain of salt; for precise mathematical statements, I recommend a good book in measure theory, such as Rudin's "Real and complex analysis", chapter on Borel measures (I think it is chapter II or III). The linked Wikipedia page also contains a bit of useful information.

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Roughly, the theorem of Riesz--Markov--Kakutani states that there is a one-one correspondence between finite Borel measures $\mu$ on $\mathbb R^d$ and bounded linear functionals $\Lambda$ on $C_c(\mathbb R^d)$, where the latter denotes the Banach space of continuous functions with compact support, equipped with the sup norm. The correspondence is given by $$\tag{1} \Lambda f = \int f\, d\mu.$$

The regularity property of $\mu$ is essential for this to be a one-one correspondence. Given a measure $\mu$, it is rather clear that (1) defines a bounded linear functional; linearity is obvious and boundedness follows from the triangle inequality $\lvert \int f\, d\mu\rvert\le \int \lvert f \rvert\, d\mu$ (replace $d\mu$ with $d\lvert\mu\rvert$ if $\mu$ is a signed measure).

However, it is not clear how to go the other way: given a functional $\Lambda$, how to uniquely define a measure $\mu$ that verifies (1)? A measure, after all, is a mapping defined on subsets of $\mathbb R^d$, whereas (1) prescribes a mapping on $C_c(\mathbb R^d)$. The regularity fills exactly this gap.

Indeed, we can use (1) to uniquely define $\mu(O)$ and $\mu(K)$ for all open $O$ and compact $K$. This is done via the lemma of Urysohn, which states that if a compact set $K$ is contained in the open set $O$ then there is $f\in C_c(\mathbb R^d)$ such that $\mathbf 1_K\le f\le \mathbf 1_O$, where $\mathbf 1_A$ denotes the function which equals $1$ on $A$ and $0$ elsewhere. Iterating this lemma appropriately, we can approximate $\mathbf 1_K$ and $\mathbf 1_O$ arbitrarily well with functions in $C_c(\mathbb R^d)$, for which (1) gives a unique prescription. In the limit we obtain a unique meaning to $\mu(O)=\int \mathbf 1_O\, d\mu$ and $\mu(K)=\int \mathbf 1_K\, d\mu$.

And now, since we required $\mu$ to be regular, we are done. Indeed, if $E$ is a measurable subset of $\mathbb R^d$ then it must be approximated arbitrarily well by open and compact sets, in the sense of $\mu$. We can therefore apply again a limiting procedure to give a unique meaning to $\mu(E)$.

The linked Wikipedia page contains an example of failed uniqueness if $\mu$ is not required to be regular.