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 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. 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 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 holds whenever E is open or when E is Borel and μ(E) < ∞ . Theorem. Let X be a locally compact Hausdorff space. For any positive linear functional ψ {displaystyle psi } on Cc(X), there is a unique regular Borel measure μ on X such that for all f in Cc(X). One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on Cc(X). 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 (non-regular) Borel measure that assigns measure 1 to any Borel set B ⊆ [ 0 , Ω ] {displaystyle Bsubseteq } if there is closed and unbounded set C ⊆ [ 0 , Ω [ {displaystyle Csubseteq [0,Omega [} with C ⊆ B {displaystyle Csubseteq B} , and assigns measure 0 to other Borel sets. (In particular the singleton {Ω} gets measure 0, contrary to the point mass measure.)