I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek.
Theorem 3.23 (Riesz-Markov-Kakutani for $\left.C_{0}(\boldsymbol{X})^{*}\right)$. Let $X$ be a locally compact Hausdorff space and $\Lambda \in C_{0}(X)^{*}$ be a continuous linear functional on the Banach space (over $\mathbb{R}$ ) of real-valued continuous functions on $X$ vanishing at infinity. Then, there exists a unique regular Borel $\sigma$-additive signed measure $\mu$ on $X$ such that
$$
\Lambda f=\int_{X} f \mathrm{~d} \mu \quad \text { for every } f \in C_{0}(X) .
$$
Moreover, we have $\|\Lambda\|=|\mu|(X)$. On the other hand, every $\mu \in \mathcal{M}(X)$ gives rise to an element $\Lambda$ of $C_{0}(X)^{*}$ via formula (3.4). Consequently, the map $\Lambda \mapsto \mu$ is an isometric isomorphism
$$
C_{0}(X)^{*} \cong \mathcal{M}(X).
$$
Previously, the author said
Definition 3.14 Let $\mu$ be a positive Borel measure on a locally compact Hausdorff space $X$. A Borel set $E \subseteq X$ is called outer regular (resp. inner regular) if
$$
\begin{gathered}
\mu(E)=\inf \{\mu(V): E \subseteq V, V \text { is open }\} \\
\text { (resp. } \mu(E)=\sup \{\mu(K): K \subseteq E, K \text { is compact }\}) .
\end{gathered}
$$
The measure $\mu$ is called regular if every Borel subset of $X$ is both outer and inner regular.
Could you elaborate on how a signed Borel measure is defined to be "regular"? I have two options in mind but I'm not sure if any of them is correct in this context of RKM theorem?
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We use the same Definition 3.14 for signed measures.
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We use Hahn decomposition to get $\mu = \mu_+ – \mu_-$ and define $\mu$ is regular if both $\mu_+$ and $\mu_-$ are regular in the sense of Definition 3.14
Best Answer
A signed measure $\mu $ (or even a complex valued measure) defined on the Borel $\sigma $-algebra of a locally compact Hausdorff space $X$ is said to be regular if the total variation measure $|\mu |$ is regular in the sense of Definition 3.14.
For signed measures this is equivalent to both $\mu _+$ and $\mu _-$, from the Hahn decomposition, being regular while, in the complex case, this is in turn equivalent to both $\Re(\mu )$ and $\Im(\mu )$ being regular signed measures.
It is interesting to observe that, if one chooses the Baire $\sigma $-algebra instead, namely the $\sigma $-algebra generated by the compact $G_\delta $-sets, then all measures that take finite values on compact sets are automatically regular, provided $X$ is $\sigma $-compact (Theorem 27 in Royden-Fitzpatrick).
This is a very compelling reason why we should replace the Borel $\sigma $-algebra by its Baire counterpart whenever doing analysis on locally compact spaces! In my opinion, there are two reasons why we dont't do it, tradition of course being one of them but, most importantly, the reason is the original sin consisting of the following thought process that must have taken place sometime in history:
Well, this looks good, except that sets of the form $f^{-1}[a,b]$ are not only closed, but also $G_\delta $, since $$ f^{-1}[a,b]= \bigcap_{n\in {\mathbb N}}f^{-1}\Big (a-\frac 1n,b+\frac1n\Big ). $$ So there is no need to take all closed sets, but only some of them!
The question becomes even more subtle because, when $X$ is compact metrizable, all closed sets are automatically $G_\delta $, so the Borel and Baire $\sigma $-algebras coincide. However, this is not so for non-metrizable compact spaces (which were probably not in the radar of the measure theory pioneers anyway).
The price we pay until today is thus to be burdened by requiring regularity of measures in theorems such as Riesz-Markov-Kakutani and many others, while we could do away with it altogether simply by choosing the right $\sigma $-algebra!