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{{Short description|Statement about linear functionals and measures}}
{{distinguish|Riesz representation theorem|Markov–Kakutani fixed-point theorem}}
{{About||more theorems that are sometimes called Riesz's theorem|Riesz theorem (disambiguation){{!}}Riesz theorem}}

In mathematics, the '''Riesz–Markov–Kakutani representation theorem''' relates [[linear functional]]s on spaces of continuous functions on a [[locally compact space]] to [[Measure (mathematics)|measures]] in measure theory.  The theorem is named for {{harvs|txt|authorlink=Frigyes Riesz|first=Frigyes |last=Riesz|year=1909}} who introduced it for [[continuous function]]s on the [[unit interval]], {{harvs|txt|authorlink=Andrey Markov, Jr.| first=Andrey |last=Markov|year=1938}} who extended the result to some non-compact spaces, and {{harvs|txt|authorlink=Shizuo Kakutani|first=Shizuo |last=Kakutani|year=1941}} who extended the result to [[compact Hausdorff space]]s.

There are many closely related variations of the theorem, as the linear functionals can be complex, real, or [[Positive linear functional|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 [[Vanish at infinity|vanishing at infinity]] or have [[compact support]], and the measures can be [[Baire measure]]s or [[Borel regular measure | regular Borel measure]]s or [[Radon measure]]s or [[signed measure]]s or [[complex measure]]s.

== The representation theorem for positive linear functionals on ''C<sub>c</sub>''(''X'') ==

The statement of the theorem for positive [[linear functionals]] on {{math|''C<sub>c</sub>''(''X'')}}, the space of [[compact space|compactly]] [[support (mathematics)|supported]] complex-valued [[continuous function]]s, is as follows:

'''Theorem''' Let {{math| ''X''}} be a [[locally compact]] [[Hausdorff space]] and <math> \psi </math> a [[positive linear functional]]  on {{math|''C''<sub>''c''</sub>(''X'')}}. Then there exists a unique positive [[Borel measure]] <math> \mu </math> on {{math|''X''}} such that{{sfn|Rudin|1987|page=40}}
:<math>\psi(f) = \int_X f(x) \, d\mu(x), \quad \forall f \in C_c(X),</math>

which has the following additional properties for some <math>\Sigma</math> containing the [[Borel set|Borel σ-algebra]] on {{math| ''X''}}:

* <math>\mu(K)<\infty</math> for every compact <math>K\subset X</math>,
* '''[[regular measure|Outer regularity]]''': <math> \mu(E) = \inf \{\mu(U): E \subseteq U, U \mbox{ open}\} </math> holds for every [[Borel set]] <math>E\in\Sigma</math>;
* '''[[inner regular measure|Inner regularity]]''': <math> \mu(E) = \sup \{\mu(K): K \subseteq E, K \mbox{ compact}\} </math> holds whenever <math>E</math> is open or when <math>E</math> is Borel and <math>\mu(E)<\infty</math>;
* <math>(X,\Sigma,\mu)</math> is a [[complete measure space]]

As such, if all open sets in {{math|''X''}} are [[σ-compact]] then <math>\mu</math> is a '''Radon measure'''.{{sfn|Rudin|1987|page=48}}

One approach to [[measure theory]] is to start with a [[Radon measure]], defined as a positive linear functional on {{math|''C''<sub>''c''</sub>(''X'')}}. This is the way adopted by [[Nicolas Bourbaki|Bourbaki]]; it does of course assume that {{math|''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 [[Regular measure|regularity]] the Borel measure need not be unique. For example, let {{math|''X''}} be the set of ordinals at most equal to the [[first uncountable ordinal]] {{math|Ω}}, with the topology generated by "[[open interval]]s". The linear functional taking a continuous function to its value at {{math|Ω}} corresponds to the regular Borel measure with a point mass at {{math|Ω}}. However it also corresponds to the (non-regular) Borel measure that assigns measure {{math|1}} to any Borel set <math>B\subseteq [0,\Omega]</math> if there is [[club set|closed and unbounded set]] <math>C\subseteq [0,\Omega[</math> with <math>C\subseteq B</math>, and assigns measure {{math|0}} to other Borel sets. (In particular the [[Singleton_(mathematics)|singleton]] <math>\{\Omega\}</math> gets measure {{math|0}}, contrary to the point mass measure.)

== The representation theorem for the continuous dual of ''C''<sub>0</sub>(''X'') ==

The following representation, also referred to as the ''Riesz–Markov theorem'', gives a concrete realisation of the [[Dual space#Continuous dual space|topological dual space]] of {{math|''C''<sub>0</sub>(''X'')}}, the set of [[continuous function]]s on {{math|''X''}} which [[vanish at infinity]]. 

'''Theorem''' Let {{math|''X''}} be a locally compact Hausdorff space.  For any continuous [[linear functional]] <math> \psi </math> on {{math|''C''<sub>0</sub>(''X'')}}, there is a unique complex-valued [[Borel regular measure|regular Borel measure]] <math>\mu</math> on {{math|''X''}} such that
:<math>\psi(f) = \int_X f(x) \, d \mu(x), \quad \forall f \in C_0(X).</math>
A complex-valued Borel measure <math>\mu</math> is called regular if the positive measure <math>| \mu |</math> satisfies the [[Regular measure|regularity]] conditions defined above. The norm of  <math> \psi </math> as a linear functional is the [[total variation]] of <math>\mu</math>, that is
:<math> \|\psi\| = |\mu|(X).</math>
Finally, <math> \psi </math> is [[positive linear functional|positive]] if and only if the measure <math>\mu</math> 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==
In its original form by {{harvs|txt|authorlink=Frigyes Riesz|first=Frigyes |last=Riesz|year=1909}} the theorem states that every continuous linear functional {{math|''A''}} over the space {{math|''C''([0, 1])}} of continuous functions {{math|''f''}} in the interval {{math|[0, 1]}} can be represented as

:<math>A[f(x)] = \int_0^1 f(x)\,d\alpha(x),</math>

where {{math|''α''(''x'')}} is a function of [[bounded variation]] on the interval {{math|[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.{{sfn|Gray|1984}} 

== Notes ==
{{Reflist}}

== References ==
* {{cite journal | last1 = Fréchet | first1 = M. | year = 1907 | title = Sur les ensembles de fonctions et les opérations linéaires | journal = [[Les Comptes rendus de l'Académie des sciences|C. R. Acad. Sci. Paris]] | volume = 144 | pages = 1414–1416 }}
* {{cite journal | last=Gray | first=J. D. | year=1984 | title=The shaping of the Riesz representation theorem: A chapter in the history of analysis | journal= Archive for History of Exact Sciences | volume=31 | issue=2 | pages=127&ndash;187 | doi=10.1007/BF00348293 }}
* {{cite journal | last=Hartig | first=Donald G. | title=The Riesz representation theorem revisited | journal=[[American Mathematical Monthly]] | volume=90 | issue=4 | pages=277&ndash;280 | doi=10.2307/2975760 | jstor=2975760 | year=1983}}; a category theoretic presentation as natural transformation.
*{{cite journal | last=Kakutani | first= Shizuo | title=Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.) | mr=0005778| journal=Ann. of Math. | series=Series 2 | volume=42 | issue=4 | year=1941 | pages= 994–1024 | doi=10.2307/1968778 | jstor=1968778 | hdl=10338.dmlcz/100940 | hdl-access=free}} 
*{{cite journal | last=Markov | first= A. | zbl= 0020.10804 | title=On mean values and exterior densities | journal= Rec. Math. Moscou | series=N.S. | volume=4 | pages= 165–190 | year=1938}} 
* {{cite journal | last1 = Riesz | first1 = F. | year = 1907 | title = Sur une espèce de géométrie analytique des systèmes de fonctions sommables | journal = C. R. Acad. Sci. Paris | volume = 144 | pages = 1409–1411 }}
* {{cite journal | last1 = Riesz | first1 = F. | year = 1909 | title = Sur les opérations fonctionnelles linéaires | journal = C. R. Acad. Sci. Paris | volume = 149 | pages = 974–977 }}
* {{cite book |last=Halmos |first=P. |title=Measure Theory |publisher=D. van Nostrand and Co. |year=1950}}
* {{mathworld|urlname=RieszRepresentationTheorem|title=Riesz Representation Theorem}}
* {{cite book|isbn=0-07-100276-6|title=Real and Complex Analysis|last1=Rudin|first1=Walter|year=1987|publisher=McGraw-Hill }}

{{Measure theory}}
{{Functional analysis}}

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[[Category:Theorems in functional analysis]]
[[Category:Duality theories]]
[[Category:Integral representations]]
[[Category:Linear functionals]]