Math Problems Vault

Problem 21.39

(i)(ii)

Let $X$ be a locally compact, Hausdorff space and $C_{0}(X)$ the space of all uniform limits of functions in $C_{c}(X)$ .

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(i)

Show that a continuous real-valued function $f$ on $X$ belongs to $C_{0}(X)$ if and only if for each $\alpha > 0$ the set $\left\{ x \in X \mid \left|f(x)\right| \geq \alpha \right\}$ is compact.

(ii)

Let $X^{*}$ be the one-point compactification of $X$ . Show that $C_{0}(X)$ consists precisely of the restrictions to $X$ of those functions in $C\left(X^{*}\right)$ that vanish at the point at infinity.

Problem 21.40

Let $X$ be an uncountable set with the discrete topology.

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(i)

What is $C_{c}(X)$ ?

(ii)

What are the Borel subsets of $X$ ?

(iii)

Let $X^{*}$ be the one-point compactification of $X$ . What is $C\left(X^{*}\right)$ ?

(iv)

What are the Borel subsets of $X^{*}$ ?

(v)

Show that there is a Borel measure $\mu$ on $X^{*}$ such that $\mu \left(X^{*}\right) = 1$ and $\int_{X} f d \mu = 0$ for each $f$ in $C_{c}(X)$ .

Problem 21.41

(i)(ii)(iii)

Let $X$ and $Y$ be two locally compact, Hausdorff spaces.

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(i)

Show that each $f \in C_{c}(X \times Y)$ is the limit of sums of the form

\sum _{i = 1}^{n} \varphi _{i}(x) \psi _{i}(y)

where $\varphi_{i} \in C_{c}(X)$ and $\psi_{i} = C_{c}(Y)$ . (The Stone-Weierstrass Theorem is useful.)

(ii)

Show that $\mathcal{B}(X \times Y) \subseteq \mathcal{B}(X) \times \mathcal{B}(Y)$ .

(iii)

Show that $\mathcal{B}(X \times Y) = \mathcal{B}(X) \times \mathcal{B}(Y)$ if and only if $X$ or $Y$ is the union of a countable collection of compact subsets.

Problem 21.42

In the proof of the Riesz-Markov Theorem, establish inner regularity in the case in which $\mu (\mathcal{O}) = \infty$ .

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Problem 21.43

(i)(ii)

Let $k(x, y)$ be a bounded Borel measurable function on $X \times Y$ , and let $\mu$ and $\nu$ be Radon measures on $X$ and $Y$ .

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(i)

Show that

\begin{aligned} \iint _{X \times Y} \varphi (x) k(x, y) \psi (y) d(\mu \times \nu ) & = \int _{Y}\left[\int _{X} \varphi (x) k(x, y) d \mu \right] \psi (y) d \nu \\ & = \int _{X} \varphi (x)\left[\int _{Y} k(x, y) \psi (y) d \nu \right] d \mu \end{aligned}

for all $\varphi \in C_{c}(X)$ and $\psi \in C_{c}(Y)$ .

(ii)

If the integral in (i) is zero for all $\varphi$ and $\psi$ in $C_{c}(X)$ and $C_{c}(Y)$ , show that then $k = 0$ almost everywhere $[\mu \times \nu ]$ .

Problem 21.44

Let $X$ be a compact, Hausdorff space and $\mu$ a Borel measure on $\mathcal{B}(X)$ . Show that there is a constant $c > 0$ such that

\left|\int _{X} f d \mu \right| \leq c\left\| f\right\| _{\max } \text{ for all } f \in C(X) .

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