View source for Riesz representation theorem

{{MSC|28A33}}

[[Category:Classical measure theory]]

{{TEX|done}}

A central theorem in classical measure theory, sometimes called
Riesz-Markov theorem, which states the following.
Let $X$ be a compact Hausdorff topological space, $C(X)$ the [[Banach space]]
of real valued continuous functions on $X$ and $L: C(X)\to \mathbb R$ a
continuous linear functional which is nonnegative, i.e. such that $L(f)\geq 0$
whenever $f\geq 0$. Then there is a [[Radon measure]] $\mu$ on the $\sigma$-algebra
of [[Borel set|Borel sets]] $\mathcal{B} (X)$ such that
\[
L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, .
\]

An analogous statement which is commonly referred to as Riesz representation theorem
is that, under the assumptions above, the dual of $C(X)$ is the space $\mathcal{M}^b (X)$
of  $\mathbb R$-valued measures with finite total variation (cp. with  [[Convergence of measures]] for the relevant definitions). Combined with  the [[Radon-Nikodým theorem]],
this amounts to the following alternative statement: for any element $L\in (C(X))'$
there are a Radon measure $\mu$ and a Borel function $g$
such that $|g|=1$ $\mu$-a.e. and
\[
L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, .
\]

More general statements for locally compact Hausdorff spaces can
be easily derived from the ones above.

The statement can be also generalized to a similar description of the dual of $C(X,B)$ when $B$ is Banach space. For instance, if $B$ is a finite-dimensional space, then for any $L\in C(X,B)'$ there is a Radon measure $\mu$ on $X$ and a Borel measurable map $g: X\to B'$ such that $\|g\|_{B'}=1$ $\mu$-a.e. and
\[
L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X, B)\, .
\]

====References====
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