The Representation of the Dual of $C(X)$: The Riesz-Kakutani Theorem

Let $x_{0}$ be a point in the compact, Hausdorff space $X$. Define $L(f) = f\left(x_{0}\right)$ for each $f \in C(X)$. Show that $L$ is a bounded linear functional on $C(X)$. Find the signed Radon measure that represents $L$.

Let $X$ be a compact, Hausdorff space and $\mu$ a Borel measure on $\mathcal{B}(X)$. Show that there is a Radon measure $\mu_{0}$ for which

Let $g_{1}$ and $g_{2}$ be two increasing functions on the closed, bounded interval $[a, b]$ that agree at the end-points. Show that

if and only if $g_{1}\left(x^{+}\right) = g_{2}\left(x^{+}\right)$for all $a \leq x < b$.

Let $X$ be a compact, Hausdorff space. Show that the Jordan Decomposition Theorem for signed Borel measures on $\mathcal{B}(X)$ follows from the Riesz-Kakutani Representation Theorem for the dual of $C(X)$ and Proposition 12.

The Riesz-Kakutani Representation Theorem: Let $X$ be a compact, Hausdorff space and $C(X)$ the linear space of continuous real-valued functions on $X$, normed by the maximum norm. Define the operator $T: \operatorname {Radon}(X) \rightarrow [C(X)]^{*}$ by setting, for $\nu \in \operatorname {Radon}(X)$, $T\nu (f) = \int_{X} f \, d\nu$ for all $f \in C(X)$. Then $T$ is a linear isometric isomorphism of $\operatorname {Radon}(X)$ onto $[C(X)]^{*}$.

Proposition 12: Let $X$ be a compact, Hausdorff space and $C(X)$ the linear space of continuous real-valued functions on $X$, normed by the maximum norm. Then for each continuous linear functional $L$ on $C(X)$, there are two positive linear functionals $L^{+}$ and $L^{-}$ on $C(X)$ for which $L = L^{+} - L^{-}$ and $\left\| L\right\| = L^{+}(1) + L^{-}(1)$.

What are the extreme points of the unit ball of the linear space of signed Radon measures $\mathcal{R}$ adon $(X)$, where $X$ is a compact, Hausdorff space?

Let $X$ be a compact metric space. On the linear space of functions $\mathcal{F}$ defined in the statement of Theorem 16, define the norm of a function to be its total variation. Show that with this norm $\mathcal{F}$ is a Banach space.

Theorem 16 (Riesz): Let $\left[a, b\right]$ be a closed, bounded interval and $\mathcal{F}$ be the collection of real-valued functions on $\left[a, b\right]$ that are of bounded variation on $\left[a, b\right]$, continuous on the right on $(a, b)$, and vanish at $a$. Then for each bounded linear functional $\psi$ on $C\left[a,b\right]$, there is a unique function $g \in \mathcal{F}$ for which $\psi (f) = \int_{a}^{b} f(x) \, dg(x)$ for all $f \in C\left[a,b\right]$.

Let $\mathcal{A}$ be an algebra of real-valued continuous functions on a compact space $X$ that separates points and contains the constants. Let $\mathcal{A}^{\perp }$ be the set of signed Radon measures on $X$ such that $\left|\mu \right|(X) \leq 1$ and $\int_{X} f d \mu = 0$ for all $f \in \mathcal{A}$.