Riesz–Markov–Kakutani representation theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures. The theorem is named for Frigyes Riesz (1909) who introduced it for continuous functions on the unit interval, Andrey Markov (1938) who extended the result to some non-compact spaces, and Shizuo Kakutani (1941) who extended the result to compact Hausdorff spaces.
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.
The following theorem represents positive linear functionals on Cc(X), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X. The Borel sets in the following statement refer to the σ-algebra generated by the open sets.
A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if
...
Wikipedia