Equivalent measures if integral of $C_b$ functions is equal

Assume that $\left(X,d\right)$ is a metric space and let $F$ be a closed subset of $X$. For $S\subset X$ and $x\in S$, define $d(x,S):=\inf\{d(x,y),y\in S\}$.

Let $O_n:=\left\{x\in X,d(x,F)<n^{-1}\right\}$. Then $O_n$ is open and the map $$f_n\colon x\mapsto \frac{d\left(x,X\setminus O_n\right)}{d\left(x,X\setminus O_n\right)+d\left(x,F\right)}$$ is continuous and bounded. It converges pointwise and monotonically to the characteristic function of $F$. So we get by monotone convergence that $\mu(F)=\nu(F)$ for all closed set $F$.

Now given a Borel set $B$ and $\varepsilon>0$, we can find a closed set $F$ and an open set $O$ such that $F\subset S\subset O$ and $\mu(O\setminus S)\leqslant\varepsilon$. Conclude.