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 in measure theory. The theorem is named for who introduced it for continuous functions on the unit interval, who extended the result to some non-compact spaces, and 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 representation theorem for positive linear functionals on ''Cc''(''X'')

The following theorem represents positive linear functionals on Cc, 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
Theorem. Let X be a locally compact Hausdorff space. For any positive linear functional on Cc, there is a unique regular Borel measure μ on X such that
One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on Cc. This is the way adopted by Bourbaki; it does of course assume that X starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.
Without the condition of regularity the Borel measure need not be unique. For example, let X be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals". The linear functional taking a continuous function to its value at Ω corresponds to the regular Borel measure with a point mass at Ω. However it also corresponds to the Borel measure that assigns measure 1 to any Borel set if there is closed and unbounded set with, and assigns measure 0 to other Borel sets.

Historical remark

In its original form by F. Riesz the theorem states that every continuous linear functional A over the space C of continuous functions in the interval can be represented in the form
where α is a function of bounded variation on the interval , and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation, the above stated theorem generalizes the original statement of F. Riesz..

The representation theorem for the continuous dual of ''C''0(''X'')

The following theorem, also referred to as the Riesz–Markov theorem, gives a concrete realisation of the topological dual space of C0, the set of continuous functions on X which vanish at infinity. The Borel sets in the statement of the theorem also refers to the σ-algebra generated by the open sets.
If μ is a complex-valued countably additive Borel measure, μ is called regular if the non-negative countably additive measure |μ| is regular as defined above.
One can deduce this statement about linear functionals from the statement about positive linear functionals by first showing that a bounded linear functional can be written as a finite linear combination of positive ones.