Any further applications of Freudenthal's 1936 Spectral Theorem? Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz Spaces I. North-Holland, 1971. The amusing thing is that the theorem is formulated eclusively in terms of partial orders, and on top of it, its proof is also in the very same terms. Yet one of the rather direct consequences of it is the spectral representation of normal operators in Hilbert spaces. Another one is the Radon-Nykodim theorem in measure theory. And to aggravate things, it can also solve some Poisson PDEs.
Does anybody know about more recent applications of that theorem ? And how about having more appreciation for the concept of partial order ?
 A: I think that some aspects of the Daniell integral and the Riesz-Markov representation theorem are other applications, however, I'm sorry that I neither can give a good reference nor can I really proof this here. I'll try to sketch the idea, maybe I'll come back later and improve it. And you might have a look at the book "Integration - a functional approach" by Klaus Bichteler (it never mentions Riesz spaces, but it gave me some of the ideas here):
For the Daniell integral, you essentially start with a Riesz subspace $L\subseteq \mathbb{R}^X$ of all  functions on a set (maybe just bounded ones?) and positive linear functionals $\psi$ on $L$ which have the property ($\star$) that $\lim_{n\to\infty} \psi(f_n) = 0$ for every monotonely decreasing sequence $(f_n)_{n\in\mathbb{N}}$ in $L$ with pointwise limit $0$. You can then complete $L$ to a Dedekind-$\sigma$-complete Riesz subspace $L^c$ of $\mathbb{R}^X$ (using the Daniell mean from Bichteler's book) and extend the linear functionals $\psi$ to the completion by using ($\star$). The Freudenthal spectral theorem now tells you that $L^c$ is essentially generated by indicator functions on a $\sigma$-algebra $\mathfrak{a}$ on $X$, which allows you to relate this approach to the Lebesgue-integral: The measure $\mu$ associated to $\psi$ is simply defined as $\mu(S) = \psi(\chi_S)$ for every $S\in\mathfrak{a}$, where $\chi_S\in L^c$ is the indicator function of $S$.
For the Riesz-Markov theorem, you start with $L=\mathcal{C}(X)$, the Riesz space of continuous real-valued functions on a compact Hausdorff space $X$, and an arbitrary $||\,\cdot\,||_\infty$-continuous linear functional $\psi\colon \mathcal{C}(X)\to\mathbb{R}$. Here $||\,\cdot\,||_\infty$ is the usual Banach-norm of $\mathcal{C}(X)$ and the $||\,\cdot\,||_\infty$-continuous linear functionals are just the order-bounded ones, which are a Riesz space themselves, so $|\psi|$ is defined and is a $||\,\cdot\,||_\infty$-continuous positive linear functional. Now one only has to show that $\lim_{n\to\infty}|\psi|(f_n)=0$ actually holds for every monotonely decreasing sequence $(f_n)_{n\in\mathbb{N}}$ in $\mathcal{C}(X)$ with pointwise limit $0$, because $X$ is compact. So we can construct the Daniell integral, which by the above gives rise to a Lebesgue integral.
PS: I'm all for more appreciation for the concept of partial order.
