# Let X be a locally comapct and Hausdorff space. We say a positive Radon Measure on X is faithful if 0<=f , int f d nu=0->f(x)=0 ∀x∈X True or false: If there is a faithful positive Radon measure on X then X has a countable dense subset ?

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

True or false: If there is a faithful positive Radon measure on $X$ then $X$ has a countable dense subset ?
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Layne Murillo
Let $\mathrm{\Gamma }$ be a set of cardinality greater than the continuum. Consider the product measures on $\left\{0,1{\right\}}^{\mathrm{\Gamma }}$ or $\left[0,1{\right]}^{\mathrm{\Gamma }}$. Then they are both faithful even though the underlying compact spaces are non-separable.
Under some extra set-theoretic assumptions, it is even possible to construct a compactification $\alpha \mathbb{N}$ of $\mathbb{N}$ with $\alpha \mathbb{N}\setminus \mathbb{N}$ non-separable having a faithful Radon measure.