The Dual of $C_c(X)$: Standing Notation

(i) Base. This setting extends Radon Measures on Locally Compact Metric Spaces: Standing Notation: its conventions and notation are in force.

(ii) The dual of $C_{c}(X)$. For a metric space $(X,d)$, $C_{c}(X)$ carries the sum, the scalar multiplication and the uniform norm $\|\cdot\|_{u}$ of The Space of Continuous Functions with Compact Support and the Uniform Norm (ii) and The Space of Continuous Functions with Compact Support and the Uniform Norm (iii), under which it is a real normed space by The Continuous Functions with Compact Support Form a Real Normed Space under the Uniform Norm (ii). $C_{c}(X)^{*}$ denotes its dual space, and $\|\Lambda\|_{*}$ the dual norm of $\Lambda\in C_{c}(X)^{*}$. Background: The Continuous Functions with Compact Support Form a Real Normed Space under the Uniform Norm and The Nonnegative Bounds of a Bounded Linear Functional Have an Infimum, Which Is a Bound.

(iii) Finite signed Borel measures. For a metric space $(X,d)$, $M(X)$ denotes the set $M(X,\mathcal{B}(X))$ of finite signed Borel measures on $X$, and $\|\nu\|$ is the total variation norm of $\nu\in M(X)$.

(iv) Order. For real functions $f,g$ on a metric space $X$, $f\le g$ means $f(x)\le g(x)$ for every $x\in X$, and $f\ge0$ means $f(x)\ge0$ for every $x\in X$; a real number $c$ in such a comparison stands for the constant function with value $c$.

(v) Use. The background facts carried by (ii), and their elementary consequences, are used without further citation.