Representation set by Riesz-Markow

This article covers the representation of positive linear forms through measures. For the representation of linear functionals in Hilbert spaces, see Fréchet-Riesz's representation theorem .

The representation theorem Riesz Markow , partly representation theorem Riesz or illustration set of Riesz Markov Kakutani called, is a mathematical theorem from the border area of the measure theory and functional analysis . It makes a statement about which positive linear forms can be represented on function spaces by means of dimensions and thus also provides descriptions of the corresponding topological dual spaces . It is named after Frigyes Riesz , Andrei Andrejewitsch Markow and Shizuo Kakutani .

motivation

If one looks at a Hausdorff space and a corresponding measure space , provided with Borel's σ-algebra and a Borel measure (in the sense of a locally finite measure ), one finds that for each , i.e. each continuous function with a compact support ${\displaystyle (X,\tau )}$ ${\displaystyle (X,{\mathcal {B}},\mu )}$ ${\displaystyle {\mathcal {B}}=\sigma (\tau )}$ ${\displaystyle f\in C_{c}(X)}$

${\displaystyle \int _{X}f\mathrm {d} \mu <\infty }$

applies. Continuous functions on a compact carrier can therefore always be integrated with respect to a Borel dimension. Also defined

${\displaystyle I:C_{c}(X)\to \mathbb {K} }$

a linear functional by

${\displaystyle I(f)=\int _{X}f\mathrm {d} \mu }$,

that is positive in the sense that

${\displaystyle f\geq 0\implies I(f)\geq 0}$

is. Building on this, the following questions arise:

1. Does a Borel measure exist for every positive functional in the sense defined above, which "represents" this functional?
2. If this Borel measure exists, is it unique ?

In addition, corresponding further questions arise: Have the above questions been answered (positively or negatively), do other topological spaces , function classes and sets of measures exist so that every positive functional can be represented by elements , and is this representation unambiguous? ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}(X)}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {F}}(X)}$ ${\displaystyle {\mathcal {M}}}$

statement

Be a Hausdorff space and the Borel σ-algebra and a Radon measure on . For so true ${\displaystyle (X,\tau )}$ ${\displaystyle {\mathcal {B}}:={\mathcal {B}}(\tau )}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \mu }$

• local finiteness : for each there is an open environment with ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle \mu (U_{x})<\infty }$
• Regularity from within : applies to everyone ${\displaystyle A\in {\mathcal {B}}}$

${\displaystyle \mu (A)=\sup\{\mu (K)\,|\,K\subset A,\,K{\text{ kompakt}}\}}$.

Furthermore, be

• ${\displaystyle C_{c}(X)}$the space for all continuous functions with a compact carrier
• ${\displaystyle C_{0}(X)}$the space of all continuous functions that vanish in infinity
• ${\displaystyle C(X)}$the space of all continuous functions .

A linear figure

${\displaystyle I:{\mathcal {F}}\to \mathbb {K} }$

of a function space is now called a positive linear form if ${\displaystyle {\mathcal {F}}}$

${\displaystyle f\geq 0\implies I(f)\geq 0}$

applies. The notation now says:

• If a space is locally compact , every positive linear form is determined by a unique Radon measure. ${\displaystyle X}$ ${\displaystyle C_{c}(X)}$
• If a space is locally compact, every positive linear form is represented by an unambiguous, finite Radon measure. ${\displaystyle X}$ ${\displaystyle C_{0}(X)}$
• If locally compact and σ-compact , every positive linear form is represented by a unique Radon measure with a compact support . ${\displaystyle X}$ ${\displaystyle C(X)}$

The representation is then given by

${\displaystyle I(f)=\int f\mathrm {d} \mu }$,

where is the corresponding (finite) Radon measure (with compact support) and the statement applies to all of the corresponding function space. ${\displaystyle \mu }$ ${\displaystyle f}$

variants

Numerous modifications to the notation set exist. So you can

• choose other topological spaces as the base space, such as completely regular spaces , ${\displaystyle X}$
• choose alternative σ-algebras such as the completion of Borel’s σ-algebra with respect to a measure or the σ-algebra of Baire sets,
• select further function classes such as the restricted continuous functions ${\displaystyle C_{b}(X)}$
• place other regularity requirements on the representing measure,

According to these various gradations, there are different ways of formulating the representation set.

Inferences

Starting from the representation of positive linear forms, the dual spaces of certain function spaces can be derived by clearly dividing a linear form into two positive linear forms ( positive part and negative part ). Sometimes these statements about the dual spaces are also called Riesz's representation theorem.

The above statements then provide that the space of the regular signed or complex measures, provided with the total variation norm, is norm isomorphic to the dual space of . ${\displaystyle C_{0}(X)}$

Web links

• Riesz representation theorem . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Riesz Representation Theorem . In: MathWorld (English).

literature

• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg / Dordrecht / London / New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin / Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .