Let X be a compact Hausdorff space and μ be a complex Borel measure on X such that ∫ X f d μ ≥ 0 for every f ∈ C ( X ) with f ≥ 0. Then show that μ is non-negative.Could anyone give some suggestion in this regard?

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Let   X be a compact Hausdorff space and   μ be a complex Borel measure on   X such that       ∫          X        f     d  μ  ≥  0 for every   f  ∈  C  (  X  ) with   f  ≥  0. Then show that   μ is non-negative.Could anyone give some suggestion in this regard?

Mateo Carson · Accepted Answer

Let   E  ⊆  X Borel. Let   ϵ  &amp;gt;  0. By regularity, we can choose a compact subset   K  ⊆  X and an open subset   U  ⊆  X with   K  ⊆  E  ⊆  U and       |    μ      |    (  U  ∖  K  )  &amp;lt;  ϵ. By Urysohn&#039;s lemma, we can pick   f  ∈  C  (  X  ,  [  0  ,  1  ]  ) with       χ    K    ≤  f  ≤      χ    U  . Then      ∫    X        |        χ    E    −  f      |    d      |    μ      |    ≤      |    μ      |    (  U  ∖  K  )  &amp;lt;  ϵ  .This shows that the function       χ    E   can be approximated by positive continuous functions in the       L    1    (      |    μ      |    )-norm.Next, let   E  ⊆  X be a Borel subset. Choose a sequence of positive continuous functions   {      f    n        }          n      =      1        ∞    ⊆  C  (  X  ,  [  0  ,  ∞  [  ) such that      ∫    X        |        f    n    −      χ    E        |    d      |    μ      |                      ⟶                    n        →        ∞              0.Hence,                              |                      ∫            X                                f            n                    d          μ          −                      ∫            X                                χ            E                    d          μ          |                ≤                  ∫          X                          |                          f          n                −                  χ          E                          |                d                  |                μ                  |                                                    ⟶                                      n              →              ∞                                      0            and we deduce that  μ  (  E  )  =      ∫    X        χ    E    d  μ  =      lim    n                                            ∫            X                                f            n                    d          μ                ⏟                    ≥      0        ≥  0.Since   E was an arbitrary Borel set, we conclude that   μ is a positive (finite) measure.

Let X be a compact Hausdorff space and μ be a complex Borel measure on X

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