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I wanna ask, if you know some interesting (or not) applications of the theorem mentioned in the title. I mean this one when we want to represent a functional as an integral over some measure. If you have some informations, proofs, exercises, books, literally everything I'll be really grateful. I am writing bachelor thesis about this theorem so I want to understand it as much as I can. Thanks in advance!

Norbert Dąbrowski
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2 Answers2

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I'm aware of three proofs of this fact. The first one is at G.Folland's great book Real Analysis: Modern Techniques and Their Applications. I have the second edition in hands and you can find the result in chapter $7$. The second proof I know is using Stone–Čech compactification and it's outlined on this post. The proof can be found in details in the book A short course on Banach space theory by N. L. Carothers, on the chapter 16. This proof has a very categorical flavor, which you may enjoy. The third proof is my favorite and can be found in the book Lectures on Functional Analysis and the Lebesgue Integral by Vilmos Komornik on the subsection 8.8 Dual Space. Riesz Representation Theorem

It's not totally related to what you want, but you may also enjoy taking a look at Daniell integral, that develops integration theory from the perspective of functionals instead of using measures.

The first reference, Folland's book, proves the result in full generality on locally compact Hausdorff spaces. The other two references prove for compact Hausdorff spaces.

Now, about applications. The first that comes to my mind is the existence of volume measure on Riemannian manifolds. If you have a Riemannian manifold $X$, then you can build the concept of integral for continuous functions, as it is usually done in courses about manifolds. Therefore, you have the linear functional $I(f) =\int_X f \mathrm{dVol}$ for all continuous functions $f$. Furthermore, you have $I(f)\geq 0$ for $f\geq 0$. Therefore, there is a measure on $X$ associated to this functional, called volume. You can also use this same trick to build Haar measures on compact Lie groups. You just build a bi-invariant Riemannian metric on the group and take the above volume. This volume will be a Haar measure, which is essentially unique by Haar's Theorem.

Hugo
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    By the way, a nice consequence of the existence of Haar's measure on the compact Lie group $SO(n)$ is the uniqueness, up to scaling, of measure on the sphere $\mathbb S^n$ invariant under action of $SO(n)$, as it is shown here. – Hugo Mar 03 '20 at 23:51
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    Another interesting application: Krylov-Bogoliubov Theorem, whose prove can be found here. The measure $\mu_n$ exists by the Riesz-Markov-Kakutani Theorem. – Hugo Mar 04 '20 at 01:18
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    Thanks a lot! Another question, do you know any well-known functional for which Riesz theorem fails? I mean on non-compact spaces of course – Norbert Dąbrowski Mar 04 '20 at 16:21
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    What about the Banach limit? – Norbert Dąbrowski Mar 04 '20 at 16:26
  • What do you mean by Banach limits? Do you mean in the categorical sense? About the first question, I don't know the answer. I'll think about it. – Hugo Mar 04 '20 at 21:45
  • Where do you expect to find a failure of the Radon-Riesz? Maybe if you drop the Hausdorff hypothesis on a compact $K$ you can find example of positive functional $I:C(K) \to R$ that does not correspond to a measure. – Hugo Mar 05 '20 at 02:33
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    @Hugo hello there! Could you elaborate more on the "existence of volume measure" application for Riemannien manifolds? – Filburt Sep 07 '23 at 08:33
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    Hi! @Filburt For instance, in general you need Riemannian volume measure in order to use $L^2$ spaces over manifolds. It is useful when investigating all sort of differential equations over manifolds. For example, when computing eigenvectors for the Laplacian or when investigating the behavior of the heat equations. The measure appears as well in order to study probability aspects over manifolds, like problems about ergodic theory/dynamical systems. Nevertheless, if you only want to integrate continuous functions, you need no measure, just the Riemannian volume form. – Hugo Sep 08 '23 at 12:05
  • @Hugo Thanks for the reply! Could you suggest a book or some notes where the definition of this measure is made this way? :-) – Filburt Sep 09 '23 at 07:02
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Here are some exercises from the time when I was TA from a functional analysis course:

  1. (A version of Prokhorov Theorem) Let $X$ be a locally compact Hausdorff space. Let $\mathcal{P}(X) \subset \mathcal{M}(X)$ be the space of probability measures. Show that $\mathcal{P}(X)$ is sequentially compact. (Here we are considering $\mathcal{M}(X)$ whit the weak-$*$ topology.)

  2. Let $X$ be a locally compact Hausdorff space and let $T:X \to X$ be a continuous map. Denote by $\mathcal{P}(M)\subset \mathcal{M}(X)$ the set of probability measures. We define the pushforward of $\mu \in \mathcal{P}(X)$ from $T$ by $T_{*}\mu(E)=\mu(T^{-1}(E))$, where $E$ is a measurable set. We say that $\mu$ is $T$-invariant if $T_{*}\mu=\mu$. Finally, define $\mathcal{M}(X,T)=\{\mu \in \mathcal{P}(X);~T_{*}\mu=\mu\}$. The idea of this problem is to show that this set is non-empty.

    2.1. Show that $$\int_{X} f d(T_{*}\mu)=\int_{X} f \circ T d\mu,$$ where $f$ be a measurable function.

    2.2. (Here is where you need to use Riesz-Markov-Kakutani Theorem) Show that $\mu \in \mathcal{M}(X,T)$ if and only if $$ \int_{X} \varphi \circ T d\mu=\int_{X} \varphi d\mu \quad \forall \varphi \in C_{c}(X).$$ 2.3. Using the Schauder Fixed Point Theorem, conclude that $\mathcal{M}(X,T) \neq \emptyset$.

  3. Let $X$ and $Y$ to be locally compact Hausdorff spaces and let $F:X \to Y$ to be a continuous sobrejective map. Let $\nu$ to be a finite measure in $Y$. Show that there exists a regular measure $\mu$ in $X$ which is Borel and satisfies $F_{*}\mu=\nu$.

Sebathon
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