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 ''C_c''(''X'')

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

Theorem Let be a locally compact Hausdorff space and $\psi$ a positive linear functional on . Then there exists a unique positive Borel measure $\mu$ on such that
:$\psi(f) = \int_X f(x) \, d\mu(x), \quad \forall f \in C_c(X),$

which has the following additional properties for some $\Sigma$ containing the Borel σ-algebra on :

* $\mu(K)<\infty$ for every compact $K\subset X$,
* Outer regularity: $\mu(E) = \inf \$ holds for every Borel set $E\in\Sigma$;
* Inner regularity: $\mu(E) = \sup \$ holds whenever $E$ is open or when $E$ is Borel and $\mu(E)<\infty$;
* $(X,\Sigma,\mu)$ is a complete measure space

As such, if all open sets in are σ-compact then $\mu$ is a Radon measure.

One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on . This is the way adopted by Bourbaki; it does of course assume that 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 be the set of ordinals at most equal to the first uncountable ordinal , with the topology generated by "open intervals". The linear functional taking a continuous function to its value at corresponds to the regular Borel measure with a point mass at . However it also corresponds to the (non-regular) Borel measure that assigns measure to any Borel set B\subseteq ,\Omega/math> if there is closed and unbounded set $C\subseteq$singleton $\$ gets measure , contrary to the point mass measure.)

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