# Why regularity is important for a Borel measure?

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 ?

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.

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.

• thank you so much sir, it was very helpful ! Commented Sep 25, 2023 at 18:29
• No problem. Glad this helped. If you think this answer is useful, upvote and accept. This is how this platform works. Commented Sep 25, 2023 at 18:41