Riesz Representation Theorem: the Dual of the Continuous Functions with Compact Support Is the Space of Finite Signed Borel Measures

In the setting of The Dual of C_c(X): Standing Notation, let $(X,d)$ be a locally compact metric space that is separable.

(i) The functional of a measure. Let $\nu\in M(X)$. The function $\Phi(\nu):C_{c}(X)\to\mathbb{R}$ given by the integral

$$\Phi(\nu)(f)=\int_{X}f\,d\nu$$

(every $f\in C_{c}(X)$ being integrable with respect to $\nu$ by Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (iii)) lies in $C_{c}(X)^{*}$, with

$$\|\Phi(\nu)\|_{*}=\|\nu\|.$$

(ii) Linearity. For $\nu,\nu'\in M(X)$ and $c\in\mathbb{R}$, and the finite signed measures $\nu+\nu'$ and $c\nu$ of Properties of the Total Variation: a Finite Measure, the Jordan Decomposition, Linear Combinations, the Total Variation Norm, and Integrability (v),

$$\Phi(\nu+\nu')=\Phi(\nu)+\Phi(\nu'),\qquad\Phi(c\nu)=c\,\Phi(\nu).$$

(iii) Representation. For every $\Lambda\in C_{c}(X)^{*}$ there is exactly one $\nu\in M(X)$ with $\Lambda=\Phi(\nu)$, that is,

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

(iv) Isomorphism. $\Phi$ is a linear isometric bijection of the real vector space $M(X)$ onto the real vector space $C_{c}(X)^{*}$:

$$\|\Phi(\nu)-\Phi(\nu')\|_{*}=\|\nu-\nu'\|\qquad(\nu,\nu'\in M(X)).$$