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

Question

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

Layne Murillo · Accepted Answer

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.

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 ?

Answered question

Answer & Explanation

New Questions in College Statistics