Generating $\sigma$-algebra and measure from the Riesz Representation Theorem

Question

Let $\Lambda:C_c(\mathbb R)\to\mathbb R$ be a positive linear functional.

The Riesz representation theorem gives that to construct the measure $\mathcal M$ and $\sigma$-algebra $\mu$, we do the following:

1. For every open set $V\subset\mathbb R$, define $$\mu(V)=\sup\{\Lambda f:f\prec V\}.$$
2. For every $E\subset\mathbb R$, define $$\mu(E)=\inf\{\mu(V):E\subset V,\ V\text{ open}\}.$$
3. Define $\mathcal M$ as the collection of all those $E\subset\mathbb R$ such that $$\mu(E)=\sup\{\mu(K):K\subset E,\ K\text{ compact}\}.$$

Here, $f\prec V$ means that $f\in C_c(\mathbb R)$, $0\leq f\leq 1$, $V$ is open, and $\operatorname{supp}(f)\subset V$.

The whole construction is eluding me.

Can someone give a specific example of a positive linear functional, and run through the steps of generating $\mathcal M$ and $\mu$?

Many thanks.

Answer 1

As copper.hat said, theorem 2.20 (and 2.14, though not very funny) of Rudin's real and complex analysis are what you're looking for.

Now i don't know what you already have learned about measures and integration theory but a good way to create measures is the following :

1) Selecta pre-measure you like. For the Lebesgue measure it will be the function defined by $\mu_0([a;b[)=b-a$ on the ring $\mathcal R$ of union of left closed right opened intervals. See wikipedia for more details.

2) Create an Outer measure $\mu^*$ from the pre-measure. This outer measure is defined on every subset of the reals as follow

$$\mu^*(A)=\inf\left\{\sum_{i=0}^\infty \mu_0(A_i) : A_i \in \mathcal R \text{ and }A \subset \bigcup_{i=0}^\infty A_i\right\}$$

For the Lebesgue case it'll gives you the Lebesgue outer measure. This method also work to create outer measures, it's used in the creation of hausdorrf measures for example.

3) Restrict that outer measure to the set $\mathcal M\subset \mathcal P(\mathbf R)$ defined by

$$\mathcal M = \left\{A\in \mathcal P(\mathbf R) : \forall B \in \mathcal P(\mathbf R), \mu^*(A\cap B)+\mu^*(A\backslash B)=\mu^*(A) \right\}.$$

It can be shown that $\mathcal M$ is a sigma algebra and that $\mu=\mu^*_{|\mathcal M}$ is a measure.

The construction from positive linear functional is just a variation of this process. You have exactly the same steps : constructing some kind of pre-measure, extending it as an outer measure with the same definition i used and finaly restricting this outer measure to a proper subset of $\mathcal P (\mathbf R)$ to obtain a measure.

Now for the specific examples : You can try for yourself to construct the measure obtained with the functional $\Lambda_\alpha(f)=f(\alpha)$ for a fixed $\alpha\in \mathbf R$.

If we take $\displaystyle\Lambda(f)=\int_{-\infty}^{+\infty}f(x) d x$ in the sense of the Riemann integral (not Lebesgue, of course) we will create the Lesbesgue measure. The combination of Urysohn's lemma and the fact that open sets of $\mathbf R^n$ can be written as the countable union of half open boxes shows that it's indeed equivalent to the pre-measure used in 1) to construct Lebesgue's measure. So it's really the volume of an open set. Once again writting any open set as a countable union of boxes "shows" (but there is a fair amount of details hidden here) that my step 2) and yours are the same : you get the Lebesgue outer measure. For step 3 you can go read the theorem 2.17 of Rudin's book. You get the Lebesgue measure and sigma algebra in step 3.