Riesz Representation Theorem for compact Hausdorff spaces I was reading about Riesz Representation Theorem for compact Hausdorff spaces $X$ on Reed & Simon's book. As far as I understood, one starts with the $\sigma$-algebra generated by $G_{\delta}$ sets (countable intersections of open sets), which is called the Baire $\sigma$-algebra and a Baire measure $\mu$ which is positive and finite measure on this $\sigma$-algebra.
We want to study the dual of $C(X)$, the space of all continuous functions $f: X \to \mathbb{C}$, which is a Banach space when equipped with the sup norm. If $f \in C(X)$ is real valued, it is automatically measurable with respect to the Baire $\sigma$-algebra; however, this is not true for general complex valued functions on $C(X)$. Thus, we ultimately we want to consider the Borel $\sigma$-algebra on $X$; it turns out that to every Baire measure on the Baire $\sigma$-algebra there exists a unique regular Borel measure on the Borel $\sigma$-algebra, so there is a natural identification between Baire measures and Borel measures.
Given a regular Borel measure $\mu$, the mapping $C(X) \ni f \mapsto \mu(f) := \int f \mu$ is a positive linear functional. Riesz Representation Theorem states that every positive linear functional is precisely of that form, so that $C(X)^{*}$ is identified with the set of regular Borel measures on $X$.
Question 1: Is this all correct?
Question 2: Can someone please point out a reference which provides the proofs of the above facts? I know this is quite standard material, but it is really hard to find more focused material. Most of the references I know prove the results for locally compact spaces, or compact metric spaces and I don't want to lose myself in all these different cases.
 A: No: complex-valued continuous functions are Baire measurable. The real and imaginary parts of a $\mathbb C$-valued continuous function are $\mathbb R$-valued continuous functions.
Note: even for compact $X$, you do not use merely $G_\delta$ sets to generate the Baire sets, but closed $G_\delta$ sets.
When the compact space $X$ is metrizable, Borel=Baire and all measures are regular.  When $X$ is not metrizable, it could happen that two different Borel measures induce the same linear functional on $C(X)$.  So, to represent the dual space $C(X)^*$, you can either (i) use Baire measures, or (ii) require regularity.
A reference:
Varadarajan, V. S., Measures on topological spaces, Am. Math. Soc., Transl., II. Ser. 48, 161-228 (1965); translation from Mat. Sb., n. Ser. 55(97), 35-100 (1961). ZBL0152.04202.
Of course it considers not only compact $X$.  For completely regular Hausdorff $X$ we can define the Baire sets as the sigma-algebra generated by the zero-sets.  [Here, "zero-sets" are sets of the form $\{x \in X : f(x) = 0\}$ where $f : X \to \mathbb R$ is continuous.]  If $X$ is compact Hausdorff, these are the same Baire sets.  But in general
$$
\{\text{compact }G_\delta\} \subsetneq \{\text{zero-sets}\} \subsetneq \{\text{closed }G_\delta\}
$$
A: This Answer is on helpful comments purely on Representation theorem part:
If you are applying Riesz representation theorem derived in Hilbert space, note that $C(X)$ is not complete w.r.t inner product $<f,g>= \int fg d\mu $. So we have to go to a completion of $C(X)$ w.r.t the norm induced by this product and apply Riesz representation in the resulting completed space (a Hilbert space). So when we apply Riesz representation theorem and write $L(f) = <f,g_L> = \int f g_L d\mu $ where $\mu$ is a borel measure, we have that $g_L$ need not be in $C(X)$ but actually in $L_2(X)$. Further by Hahn banach theorem we can extend $L$  to apply also in the completion $L_2(X)$. So you can also identify the dual space $C^*(X)$ with positive functions in $L_2(X)$. But for above to apply we need $L$ to be of bounded norm when applied on $C(X)$ w.r.t norm on $C(X)$ induced by this inner product.
EDIT:
There is another version of Riesz theorem w.r.t measure theory which applies to this case. See:
https://en.m.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem#:~:text=In%20mathematics%2C%20the%20Riesz%E2%80%93Markov,to%20measures%20in%20measure%20theory.
This theorem shows that if $L(f) \leq L(g)$ for any $f,g$ such that $f(x) \leq g(x)$, $\forall x \in X$ then such an $L$ can be represented as $L(f) = \int f d\mu_R$ for a unique Radon Measure $\mu_R$. Hence this applies for any such monotonic Linear functional $L$. If we have a decomposition that $L=L_1 - L_2$ with $L_1,L_2$ positive Linear functional then a form of Riesz Representation theorem holds for this $L$ on $C(X)$ with $X$ being compact Hausdorff space i.e., $L = \int f d\mu_{R}-\int f d\mu'_{R}$. I find such a decomposition interesting.
This extends to Locally Compact Hausdorff space $X$ but now we can represent only linear functionals on $C_c(X)$.
https://math.berkeley.edu/~arveson/Dvi/rieszMarkov.pdf
Can we extend $C_c(X)$ to $C(X)$ for $X$ being Locally compact Hausdorff space ?
