A Probability Measure on Euclidean Space Is Determined by the Integrals of Lipschitz Functions with Values in the Unit Interval

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, fix a dimension $q$. A function $\phi:\mathbb{R}^{q}\to\mathbb{R}$ is Lipschitz when it is so as a map from $\mathbb{R}^{q}$ with the Euclidean distance $d_{E}$ of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (ii) to the real line with the absolute-value metric; such a $\phi$ is uniformly continuous by A Lipschitz Map is Uniformly Continuous and Continuous, hence continuous by A Uniformly Continuous Map Between Metric Spaces Is Continuous, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii), and the functions $\phi$ considered below, taking values in $[0,1]$, are bounded, hence integrable with respect to every probability measure by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). Let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{q})$ satisfy

$$\int_{\mathbb{R}^{q}}\phi\,d\mu=\int_{\mathbb{R}^{q}}\phi\,d\nu$$

for every Lipschitz $\phi:\mathbb{R}^{q}\to\mathbb{R}$ with $0\le\phi(x)\le1$ for all $x\in\mathbb{R}^{q}$. Then $\mu=\nu$.