The Riesz–Markov Representation Theorem on a Locally Compact Metric Space

In the setting of The Dual of C_c(X): Standing Notation, let $(X,d)$ be a locally compact metric space and let $\Lambda$ be a positive linear functional on $C_{c}(X)$.

(i) Existence. There is a Radon measure $\mu$ on $X$ with

$$\Lambda f=\int_{X}f\,d\mu\qquad(f\in C_{c}(X)),$$

the integrals existing by Radon Measures and Finite Signed Borel Measures Are Determined by Their Integrals of Compactly Supported Continuous Functions (i).

(ii) Uniqueness. If $\mu$ and $\mu'$ are Radon measures on $X$ that both satisfy the identity of (i), then $\mu(B)=\mu'(B)$ for every $B\in\mathcal{B}(X)$.

(iii) Bounded functionals. If moreover $\Lambda$ is bounded, that is $\Lambda\in C_{c}(X)^{*}$, then the Radon measure $\mu$ of (i) is finite, and

$$\mu(X)=\sup\{\Lambda f:f\in C_{c}(X),\ 0\le f\le1\}=\|\Lambda\|_{*}.$$