Riesz representation theorem

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2020 Mathematics Subject Classification: Primary: 28A33 Secondary: 28A25 [MSN][ZBL]

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 sets $\mathcal{B} (X)$ such that \[ L (f) = \int_X f\, d\mu \qquad \forall f\in C (X)\, . \] The theorem is often stated in a more general version for locally compact Hausdorff spaces $X$, of which the statement above is a simple corollary (cp, with Section 2.14 of [Ru])

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 Signed measure 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 \[ L (f) = \int_X fg\, d\mu\qquad \forall f\in C(X)\, . \]

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 \[ L (f) = \int_X g (f)\, d\mu \qquad \forall f\in C(X, B)\, . \]


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