Let X be a locally compact Hausdorff space and C b (X) the set of all continuous functions with support compact. The dual of C b is isometric isomorphism of M(X), with M(X) the set of all regular Borel measures on X? Do the measures need to be finite as well?

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Let X be a locally compact Hausdorff space and       C          b      (X) the set of all continuous functions with support compact. The dual of       C          b       is isometric isomorphism of M(X), with M(X) the set of all regular Borel measures on X? Do the measures need to be finite as well?

Zayden Wiley · Accepted Answer

No, the measures in the dual space of       C    c    (  X  ) are not necessarily finite. Think of   X  =      R   and   λ the Lebesgue measure on       R  . Then  λ  (  f  )  =      ∫          R        f  (  x  )         d    λ  (  x  )defines a functional on       C    c    (      R    ), the space of continuous functions on       R   with compact support.If   X is compact, then the measures in the dual space of       C    c    (      R    ) are finite because they are regular.

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