I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. I have no question here. This thread is to summarize $3$ versions of the theorem (in an increasing order of generality).
I try to include related definitions to remove any ambiguity, but It's likely that there are subtle mistakes that I could not recognize. I'm very happy to receive your suggestions.
Let
- $\mathbb K \in \{\mathbb R, \mathbb C\}$.
- $X$ be a topological space,
- $\mathcal B$ the Borel $\sigma$-algebra of $X$
- $E :=C_c (X)$ the space of $\mathbb K$-valued compactly supported continuous functions on $X$. The support of a map $f:X \to \mathbb K$ is $\operatorname{supp} f := \overline{\{x \in X \mid f(x) \neq 0\}}$.
- If $\mathbb K = \mathbb R$ then a linear functional on $E$ is real-valued and $\mathbb R$-linear. If $\mathbb K = \mathbb C$ then a linear functional on $E$ is complex-valued and $\mathbb C$-linear.
Theorem 3.16 Let $X$ be locally compact Hausdorff and $\Lambda$ a linear (not necessarily continuous) functional on $E$ which is positive, i.e., $\Lambda f \ge 0$ for all $f \in E$ such that $f \ge 0$. Then there exist a $\sigma$-algebra $\mathfrak{M}$ on $X$ such that $\mathcal B \subset \mathfrak M$, and a non-negative (not necessarily finite) measure $\mu$ on $\mathfrak{M}$ such that
- (a) $\Lambda f=\int_{X} f \mathrm{~d} \mu$ for every $f \in E$;
- (b) for every compact set $K \subset X$ we have $\mu(K)<\infty$;
- (c) $\mu$ is outer regular on for every $B \in \mathfrak{M}$, i.e., $\mu(B) = \inf \{\mu(U) \mid B \subset U, U \text{ open}\}$.
- (d) $\mu$ is tight on every open set $B$ and every $B \in \mathfrak{M}$ with $\mu(B)<\infty$, i.e., $\mu(B) = \sup \{\mu(K) \mid K \subset B, K \text{ compact}\}$.
- (e) $(X, \mathfrak{M}, \mu)$ is complete, i.e. if $E \in \mathfrak{M}, \mu(E)=0$ and $A \subset E$, then $A \in \mathfrak{M}$.
Moreover, the measure $\mu$ is unique in the class of non-negative measures on $\mathfrak{M}$ satisfying conditions (a)-(d).
Definition 3.20. Let $\mu:\mathfrak M \to \mathbb C$ be a complex measure defined on a $\sigma$-algebra $\mathfrak{M}$ of subsets of a set $X$. For any $B \in \mathfrak{M}$ we denote by $\Pi(B)$ the collection of all measurable finite partitions of $B$, i.e., $$ \Pi(B)=\left\{\left(B_{1}, \ldots, B_{n}\right) \,\middle\vert\, n \in \mathbb{N^*}, B_{i} \in \mathfrak{M}, B_{i} \cap B_{j}=\varnothing \text { for } 1 \leq i \neq j \leq n, \bigcup_{i=1}^{n} B_{i}=B\right\} . $$ The variation $|\mu|$ of $\mu$ is defined by $$ |\mu|(B)=\sup \left\{\sum_{i=1}^{n}\left|\mu\left(B_{i}\right)\right| \,\middle\vert\, \left(B_{1}, \ldots, B_{n}\right) \in \Pi(B)\right\} \quad \forall B \in \mathfrak{M}. $$ The value $|\mu|(X)$ is called the total variation of $\mu$.
Proposition 6.1. For any complex measure $\mu$, its variation $|\mu|$ is a non-negative finite measure.
A complex Borel measure $\mu$ is called regular if its variation $|\mu|$ is both tight and outer regular on every Borel set.
Definition A signed (or real) measure is a measure that takes values in $\mathbb R$. This implies a signed measure is not allowed take the values $\pm \infty$.
Let
- $X$ be a topological space,
- $\mathcal M(X)$ the space of regular signed Borel measures on $X$.
- $E :=C_0 (X)$ the space of $\mathbb R$-valued continuous functions on $X$ vanishing at infinity.
- $E^*$ the continuous dual of $E$. Then $\Lambda \in E^*$ is real-valued and $\mathbb{R}$-linear.
Theorem 3.23 Let $X$ be locally compact Hausdorff and $\Lambda \in E^*$ . Then there exists a unique regular Borel signed measure $\mu$ on $X$ such that $$ \Lambda f=\int_{X} f \mathrm{~d} \mu \quad \text {for every} \quad f \in E \quad (\star). $$ Moreover, we have $\|\Lambda\|_{E^*} = |\mu|(X)$. On the other hand, every $\mu \in \mathcal{M}(X)$ gives rise to an element $\Lambda \in E^*$ via formula $(\star)$. Consequently, the map $\Lambda \mapsto \mu$ is an isometric isomorphism $$ E^* \cong \mathcal{M}(X). $$
Let
- $X$ be a topological space,
- $\mathcal M(X)$ the space of regular complex Borel measures on $X$.
- $E :=C_0 (X)$ the space of $\mathbb C$-valued continuous functions on $X$ vanishing at infinity.
- $E^*$ the continuous dual of $E$. Then $\Lambda \in E^*$ is complex-valued and $\mathbb{C}$-linear.
Theorem 6.10 Let $X$ be locally compact Hausdorff and $\Lambda \in E^*$. Then there exists a unique regular Borel complex measure $\mu$ on $X$ such that $$ \Lambda f=\int_{X} f \mathrm{~d} \mu \quad \text {for every} \quad f \in E \quad (\star). $$ Moreover, we have $\|\Lambda\|_{E^*} = |\mu|(X)$. On the other hand, every $\mu \in \mathcal{M}(X)$ gives rise to an element $\Lambda \in E^*$ via formula $(\star)$. Consequently, the map $\Lambda \mapsto \mu$ is an isometric isomorphism $$ E^* \cong \mathcal{M}(X). $$
