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 ?

Eliza Gregory 2022-10-20 Answered
Let X be a locally comapct and Hausdorff space. We say a positive Radon Measure on X is faithful if
0 f       ,       f d μ = 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 ?
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Answers (1)

Layne Murillo
Answered 2022-10-21 Author has 14 answers
Let Γ be a set of cardinality greater than the continuum. Consider the product measures on { 0 , 1 } Γ or [ 0 , 1 ] Γ . 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 α N of N with α N N non-separable having a faithful Radon measure.
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