The linear functional taking a continuous function to its value at ? corresponds to the regular Borel measure with a point mass at ?.
2.
In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.
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A current such that "'M "'( " T " ) < " is representable by integration of a regular Borel measure by a version of the Riesz representation theorem.
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Further generalisations are sometimes considered, for example A ^ 2 _ \ nu denotes a weighted Bergman space ( often called a Zen space ) with respect to a translation-invariant positive regular Borel measure \ nu on the closed right complex half-plane \ overline { \ mathbb { C } _ + }, that is
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There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures.
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