# Proofs of the Riesz–Markov–Kakutani representation theorem

Let $X$ be a compact Hausdorff space, $C(X)$ the set of all real continuous functions on $X$, and $\mathcal{B}$ be the Baire $\sigma$-algebra of $X$, which is the $\sigma$-algebra generated by the functions in $C(X)$. Furthermore $M(X)$ is the set of all finite signed measures on $(X,\mathcal{B})$ with the norm $||\mu||=|\mu|(X)$, $|\mu|$ being the total variation of $\mu$. Finally, for all $\mu \in M(X)$ define $\phi_{\mu}:C(X)\rightarrow \mathbb{R}$ by $\phi_{\mu}(f)=\int f d\mu$.

It is possible to formulate the Riesz-Markov-Kakutani theorem as follows:

The application $\mu\mapsto \phi_{\mu}$ is a surjective isometry from $M(X),||.||$ onto $C(X)^{*},||.||_{*}$ the dual space of $C(X),||.||_{\infty}$.

In other words the topological dual of the Banach space $(C(X),||.||_{\infty})$ (with as usual $||f||_{\infty}=\sup_{x\in X} f(x))$) can be identified with $M(X)$.

My question is: since there is a functional analysis formulation of the theorem, is there a functional-analysis-flavoured proof of it ?

Note that this version of the Riesz-Markov-Kakutani theorem is much stronger than the usually stated one, which is concerned positive functionals on $\mathbb{R}$. The fact that the dual norm is the total variation one is equivalent to the fact that Baire measures are necessarily regular, a not so trivial fact proved in Halmos's Measure Theory.

I find proofs like the one in Rudin's Real and complex analysis disturbingly artificial and complex for such what seems such a natural and important result: an integral is nothing more or less than a bounded linear functional on continuous functions (when integrating over a compact space).

I had a professor who used Daniell's integral with which the Riesz-Markov-Kakutani theorem follows almost instantly. That is a very good proof, the best I've seen so far, but I'm still under the impression that there might be other ways to look at the problem.

My favourite proof is due to Hartig:

D. G. Hartig, The Riesz representation theorem revisited, American Mathematical Monthly, 90(4), 277–280.

Hartig claims that his proof is ...category-theoretic but having unwrapped the details I must say it is really functional-analytic. The idea is as follows.

• Step 1. We can easily prove this theorem for extremely disconnected compact Hausdorff spaces. Indeed, everything can be reduced there to messing around with indicator functions of clopen subsets.

• Step 2. We employ the Stone–Čech compactification of a discrete space. One can use the Banach–Alaoglu theorem to show that every completely regular space admits the Stone–Čech compactification. The Stone–Čech compactification of a discrete space is extremely disconnected.

• Step 3. Now, we prove that every compact space $X$ is a continuous image of an extremely disconnected space. Indeed, give $X$ the discrete topology. Call this space $X_d$ and extend the identity map $\iota \colon X_d \to X$ to a continuous map from $\beta X_d$ to $X$. Then, transfer the Riesz theorem to $X$ via the adjoint map to $f\mapsto f\circ (\beta\iota)$ ($f\in C(X)$).

• Thanks a lot ! I'm not learned enough to understand everything you said but I'm quite sure it's the kind of answer I'm looking for. I'll be studying categories this summer and I'm looking forward to reading Hartig's paper ! – Sergio May 6 '15 at 22:29
• The proof is due to Garling, Hartig is only giving an exposition. – Michael Greinecker Jan 7 '17 at 0:32
• Thanks Tomasz for the nice exposition!! :) – C-Star-W-Star Jan 14 '18 at 14:10
• @FreeziiS, Alex, you are most welcome :-) – Tomek Kania Jan 14 '18 at 15:09