Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if

$$0\le f\text{}\text{}\text{},\text{}\text{}\text{}\int fd\mu =0\to f(x)=0\text{}\text{}\mathrm{\forall}x\in X$$

True or false: If there is a faithful positive Radon measure on $X$ then $X$ has a countable dense subset ?

$$0\le f\text{}\text{}\text{},\text{}\text{}\text{}\int fd\mu =0\to f(x)=0\text{}\text{}\mathrm{\forall}x\in X$$

True or false: If there is a faithful positive Radon measure on $X$ then $X$ has a countable dense subset ?