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.

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)

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The statement of the theorem for positive linear functionals on Cc(X), the space of compactly supported complex-valued continuous functions, is as follows:

Theorem Let X be a locally compact Hausdorff space and   a positive linear functional on Cc(X). Then there exists a unique positive Borel measure   on X such that[1]

 

which has the following additional properties for some   containing the Borel σ-algebra on X:

  •   for every compact  ,
  • Outer regularity:   holds for every Borel set  ;
  • Inner regularity:   holds whenever   is open or when   is Borel and  ;
  •   is a complete measure space

As such, if all open sets in X are σ-compact then   is a Radon measure.[2]

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   if there is closed and unbounded set   with  , and assigns measure 0 to other Borel sets. (In particular the singleton   gets measure 0, contrary to the point mass measure.)

The representation theorem for the continuous dual of C0(X)

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The following representation, also referred to as the Riesz–Markov theorem, gives a concrete realisation of the topological dual space of C0(X), the set of continuous functions on X which vanish at infinity.

Theorem Let X be a locally compact Hausdorff space. For any continuous linear functional   on C0(X), there is a unique complex-valued regular Borel measure   on X such that

 

A complex-valued Borel measure   is called regular if the positive measure   satisfies the regularity conditions defined above. The norm of   as a linear functional is the total variation of  , that is

 

Finally,   is positive if and only if the measure   is positive.

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.

Historical remark

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In its original form by Frigyes Riesz (1909) the theorem states that every continuous linear functional A over the space C([0, 1]) of continuous functions f in the interval [0, 1] can be represented as

 

where α(x) is a function of bounded variation on the interval [0, 1], 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 (that assigns to each function of bounded variation the corresponding Lebesgue–Stieltjes measure, and the integral with respect to the Lebesgue–Stieltjes measure agrees with the Riemann–Stieltjes integral for continuous functions), the above stated theorem generalizes the original statement of F. Riesz.[3]

Notes

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  1. ^ Rudin 1987, p. 40.
  2. ^ Rudin 1987, p. 48.
  3. ^ Gray 1984.

References

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  • Fréchet, M. (1907). "Sur les ensembles de fonctions et les opérations linéaires". C. R. Acad. Sci. Paris. 144: 1414–1416.
  • Gray, J. D. (1984). "The shaping of the Riesz representation theorem: A chapter in the history of analysis". Archive for History of Exact Sciences. 31 (2): 127–187. doi:10.1007/BF00348293.
  • Hartig, Donald G. (1983). "The Riesz representation theorem revisited". American Mathematical Monthly. 90 (4): 277–280. doi:10.2307/2975760. JSTOR 2975760.; a category theoretic presentation as natural transformation.
  • Kakutani, Shizuo (1941). "Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.)". Ann. of Math. Series 2. 42 (4): 994–1024. doi:10.2307/1968778. hdl:10338.dmlcz/100940. JSTOR 1968778. MR 0005778.
  • Markov, A. (1938). "On mean values and exterior densities". Rec. Math. Moscou. N.S. 4: 165–190. Zbl 0020.10804.
  • Riesz, F. (1907). "Sur une espèce de géométrie analytique des systèmes de fonctions sommables". C. R. Acad. Sci. Paris. 144: 1409–1411.
  • Riesz, F. (1909). "Sur les opérations fonctionnelles linéaires". C. R. Acad. Sci. Paris. 149: 974–977.
  • Halmos, P. (1950). Measure Theory. D. van Nostrand and Co.
  • Weisstein, Eric W. "Riesz Representation Theorem". MathWorld.
  • Rudin, Walter (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-100276-6.