What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

Question

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued linear functional, $\lambda$ defined on the set of continuous functions with compact support in a "nice" topological space (Hausdorff and locally compact) the theorem states there is a corresponding unique positive complete measure $\mu$ defined on a sigma algebra containing all borel sets in X so that the integral of $f$ w.r.t $\mu$, $\int_{X}{f}d\mu$= $\lambda(f)$.

There are a few other parts of the theorem's statement, but I stop here to ask: "What are some examples of corresponding measures to linear functionals that are not immediately obviously translatable to a measure without the theorem?"

An example I thought of first was a linear functional that gives the n'th coefficient of some function $f$ represented by a polynomial, possibly trigonometric. But I also wondered how differentiation at a point would correspond to a measure. Are these questions well-posed? If so could you help me think about them?

Answer 1

The spectral theorem for bounded selfadjoint operators is a surprising consequence of the Riesz Representation Theorem for continuous functions $C[a,b]$ on a finite interval $[a,b]\subset\mathbb{R}$.

If $A$ is a bounded selfadjoint operator on a Hilbert space $X$ with spectrum $\sigma(A)$, it is not so difficult to show that $\|p(A)\| \le \sup_{\lambda\in\sigma(A)}|p(\lambda)|$ for polynomials $p$. This is because the norm and spectral radius of a selfadjoint are found to be the same. Therefore, the map $p\mapsto p(A)$ then extends to continuous functions $p$ on $\sigma(A)$, and gives the existence of unique finite complex Borel measures $\mu_{x,y}$ such that $$ (p(A)x,y) = \int_{\sigma(A)}p(\lambda)d\mu_{x,y}(\lambda). $$ And $\mu_{x,x}$ is a positive Borel measure with $\|x\|^{2}=\mu_{x,x}(\sigma(A))$. For a fixed Borel subset of $\sigma(A)$, this leads to a bounded sesquilinear form $\mu_{x,y}(S)$ with $|\mu_{x,y}(S)|\le \|x\|\|y\|$ and gives the existence of a unique positive Borel operator measure $E$ such that $$ \mu_{x,y}(S) = (E(S)x,y). $$ After a little study, one finds that $E$ is a spectral measure, i.e., satisfies $$ E(\sigma(A))=I,\;\; E(S)^{\star}=E(S)=E(S)^{2},\;\; E(S)E(T)=E(S\cap T). $$ This then yields the spectral theorem $$ A = \int_{\sigma(A)}\lambda dE(\lambda). $$ For every polynomial $p(\lambda)$, one automatically has $$ p(A) = \int_{\sigma(A)}p(\lambda)\,dE(\lambda), $$ which extends to continuous functions $p$, and satisfies $$ p(A)q(A) = (pq)(A). $$

In this light, the Spectral Theorem for selfadjoint linear operators is a lifting of the Riesz Representation to an operator measure $E$ representation.

Answer 2

My favorite example is harmonic measure. Let $\Omega$ be a domain in $\mathbb R^n$ (with smooth boundary $\Gamma=\partial \Omega$, to avoid technicalities). For every $\phi\in C(\Gamma)$ there is a unique harmonic function $u$ on $\Omega$ such that $u(x)\to \phi(y)$ as $x\to y$, for every $y\in \partial\Omega$.

Fix $z\in\Omega$. By the maximum principle, $|u(z)|\le \max_\Gamma|\phi|$. Thus, the map $\phi\mapsto u(z)$ is a bounded linear functional on $C(\Gamma)$. By Riesz representation, there is a measure $\omega_z$ on $\Gamma$ such that $$u(z)= \int_{\Gamma} \phi(y)\, d\omega_z(y)$$ (Note that $\omega_z$ is a probability measure.)

While for smooth domains one can relate $\omega_z$ to a familiar object (Green's function), for general domains, where $\omega_z$ still makes sense, its structure is opaque. For $n>2$, it is unknown how large the support of $\omega_z$ can be: that is, what is the smallest $d=d(n)$ such that every $\omega_z$ gives mass $1$ to some set of Hausdorff dimension $d$? In 1980s Bourgain proved $d(n)<n$; in 1990s Wolff proved that $d(3)>2$... and I think this is where it remains now.

Answer 3

My favourite examples are the Riesz products, that is, weak*-limits of

$$\prod_{k=1}^n \big(1+\cos(3^kt)\big)$$

in $C(\mathbb{T})^*$. Colin C. Graham used the Riesz products to give a slick proof of the Wiener–Pitt phenomenon in $M(\mathbb{T})$.

Answer 4

A good example is the average of $f\in C[0,1]$ on the Cantor set.

If $C=\bigcap_{n\in\mathbb N}C_n$, where $C_1=[0,1/3]\cup [2/3,1]$, $C_2=[0,1/9]\cup[2/9,1/3]\cup[6/9,7/9]\cup[8/9,1]$, etc.

As $\mu(C_n)=2^n/3^n$, we define $$ \varphi_n(f)=\frac{1}{m(C_n)}\int_{C_n}f(x)\,dx, $$ and show that the sequence $\varphi_n(f)$ converges, for every such $f$, to a limit $\varphi(f)$, with $\lvert\varphi(f)\rvert\le \|f\|_\infty$. This defines a weak$^*$ convergence of $\varphi_n\to\varphi$ which is a unit measure, singular to Lebesgue measure and containing no atomic (Dirac) measures.