Let $X$ be a compact Hausdorff topological house, and $mathcal C^0 (X) = {f:Xtomathbb{R}; f textual content{ is steady }}$. It’s well-known that for any bounded linear practical $phi: mathcal C^0(X)tomathbb{R},$ such that $phi(f)geq 0$ if $fgeq 0$ ($phi$ is known as a optimistic linear practical), then there exists a singular common Borel measure $mu$, such that

$$phi(g) = int g mathrm dmu, forall gin mathcal C^0(X). $$

This end result follows from a direct utility of Riesz–Markov–Kakutani illustration theorem.

If we drop the Hausdorff speculation (solely assuming $X$ as compact topological house). Then we are able to lose the individuality of the measure that represents the linear practical. A well-known instance is the compact topological house “$[0,1]$ with to origins”. On this case the practical $phi: mathcal C^0(X)tomathbb{R}$, $phi(f) = f(0)$ could be written as $int f mathrm{d}delta_0$ or $int f mathrm{d}delta_{0′}.$

I wish to know if we nonetheless have the existence of a measure that represents the practical. In different phrases, I wish to know if the next theorem is true

Doable Theorem:Let $(X,tau)$ be a compact non-Hausdorff house, and $Lambda : mathcal C^0(X)tomathbb{R}$ a optimistic bounded linear practical, then there exists a measure $mu: mathcal B(tau)to mathbb{R}$ (the place $mathcal B(tau)$ is the smallest $sigma$-algebra such that $tausubset mathcal B(tau))$, such that

$$Lambda(f) = int f mathrm{d}mu, forall fin mathcal C^0(X).$$

Can anybody assist me?

I’ve searched on-line however I used to be not capable of finding a end result within the non-Hausdorff case.