Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Question

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^0(X)\to\mathbb{R},$ such that $\phi(f)\geq 0$ if $f\geq 0$ ($\phi$ is called a positive linear functional), then there exists a unique regular Borel measure $\mu$, such that $$\phi(g) = \int g\ \mathrm d\mu, \ \forall \ g\in \mathcal C^0(X). $$ This result follows from a direct application of Riesz–Markov–Kakutani representation theorem.

If we drop the Hausdorff hypothesis (only assuming $X$ as compact topological space). Then we can lose the uniqueness of the measure that represents the linear functional. A famous example is the compact topological space "$[0,1]$ with to origins". In this case the functional $\phi: \mathcal C^0(X)\to\mathbb{R}$, $\phi(f) = f(0)$ can be written as $\int f\ \mathrm{d}\delta_0$ or $\int f\ \mathrm{d}\delta_{0'}.$

I would like to know if we still have the existence of a measure that represents the functional. In other words, I would like to know if the following theorem is true

Possible Theorem: Let $(X,\tau)$ be a compact non-Hausdorff space, and $\Lambda : \mathcal C^0(X)\to\mathbb{R}$ a positive bounded linear functional, then there exists a measure $\mu: \mathcal B(\tau)\to \mathbb{R}$ (where $\mathcal B(\tau)$ is the smallest $\sigma$-algebra such that $\tau\subset \mathcal B(\tau))$, such that $$\Lambda(f) = \int f\ \mathrm{d}\mu, \ \forall \ f\in \mathcal C^0(X).$$

Can anyone help me?

I have searched online but I was not able to find a result in the non-Hausdorff case.

Answer 1

The answer is yes.

First, it follows from the following result and the Riesz–Markov–Kakutani representation theorem that we can always find a suitable Baire measure representing a positive linear functional.

Theorem: Let $X$ be any topological space. Then there exists a completely regular Hausdorff space $Y$ and a continuous surjection $\tau:X\to Y$ such that the function $g\mapsto g\circ\tau$ is an isomorphism from $C_B(Y)$ onto $C_B(X)$.

This is Theorem 3.9 of "Rings of continuous functions" (1960) by Gillman and Jerison.

So the problem reduces to the question whether a Baire measure on a compact topological space can be extended to a Borel measure. We can do this using the following result, which specializes the very abstract Theorem 2.6.1 of "Convex Cones" (1981) by Fuchssteiner and Lusky.

Theorem: Let $X$ be a non-empty compact topological space and $L:\mathcal{C}^0_+(X)\to\mathbb{R}$ be an additive function on the cone of nonnegative continuous functions on $X$ such that $L(g)\leq\max g$ for all $g$. Then there exists a Borel probability measure $\nu$ on $X$ such that $$L(g)\leq\int g~\mathrm d\nu$$ for all $g\in \mathcal{C}^0_+(X)$.

For nonzero $\Lambda$, let $L=1/\Lambda(1)\cdot \Lambda$. Then the measure $\mu=\Lambda(1)\cdot\nu$ does the trick.

It should be noted that the resulting Borel measure need not be regular. For non-Hausdorff $X$, there is no point in going beyond Baire measures.