Radon Measures and Finite Signed Borel Measures Are Determined by Their Integrals of Compactly Supported Continuous Functions

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,d)$ be a locally compact metric space.

(i) Integrability. Every function $f$ in $C_{c}(X)$ is integrable with respect to every Radon measure on $X$, and integrable with respect to every finite signed Radon measure on $X$.

(ii) Radon measures. Let $\mu$ and $\mu'$ be Radon measures on $X$ with

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

Then $\mu(B)=\mu'(B)$ for every $B\in\mathcal{B}(X)$.

(iii) Finite signed Borel measures. Let $X$ moreover be separable, and let $\nu$ and $\nu'$ be finite signed Borel measures on $X$ whose integrals agree on $C_{c}(X)$:

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

Then $\nu(B)=\nu'(B)$ for every $B\in\mathcal{B}(X)$.