Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits

Let $(X,d)$ be a metric space with $X$ nonempty and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. Regard $\mathbb{R}$ as a metric space with the absolute-value metric, and let bounded have the meaning fixed there for a real-valued function on a set.

1. (Determination) Let $\mu$ and $\nu$ be finite measures on $(X,\mathcal{B}(X))$ such that

$$\int_X f\,d\mu=\int_X f\,d\nu$$

for every bounded Lipschitz function $f:X\to\mathbb{R}$, the integrals being those of Integrable Function and the Lebesgue Integral. Then $\mu=\nu$, that is, $\mu(B)=\nu(B)$ for every $B\in\mathcal{B}(X)$.

2. (Uniqueness of weak limits) Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of finite measures on $(X,\mathcal{B}(X))$ and let $\mu$ and $\nu$ be finite measures on $(X,\mathcal{B}(X))$ such that $(\mu_n)_{n\in\mathbb{N}}$ converges weakly both to $\mu$ and to $\nu$. Then $\mu=\nu$.