Measure and Topology

Let $X$ be a locally compact, Hausdorff space, and $F$ a set that has closed intersection with each compact subset of $X$. Show that $F$ is closed.

Regarding the proof of Proposition 3:

Proposition 3: Let $F$ be a compact subset of a locally compact, Hausdorff space $X$ and $O$ a neighborhood of $F$. Then there is a function $f \in C_{c}(X)$ for which $f = 1$ on $F$, $f = 0$ on $X \sim O$ and $0 \leq f \leq 1$ on $X$. The proof chooses (by the preceding proposition) a neighborhood $\mathcal{V}$ of $F$ that has compact closure contained in $O$; since $\overline{\mathcal{V}}$ is compact and Hausdorff, it is normal, and $F$ and $\overline{\mathcal{V}} \sim \mathcal{V}$ are disjoint closed subsets of $\overline{\mathcal{V}}$; Urysohn's Lemma then gives a continuous function $f$ on $\overline{\mathcal{V}}$ with $f = 1$ on $F$, $f = 0$ on $\overline{\mathcal{V}} \sim \mathcal{V}$ and $0 \leq f \leq 1$ on $\overline{\mathcal{V}}$, extended to all of $X$ by setting $f = 0$ on $X \sim \overline{\mathcal{V}}$.

Let $X$ be a locally compact, Hausdorff space and $X^{*}$ the Alexandroff one-point compactification of $X$ :

Show that the Alexandroff one-point compactification of $\mathbb {R}^{n}$ is homeomorphic to the $n$-sphere $S^{n} = \left\{ x \in \mathbb {R}^{n+1} \mid \left\| x\right\| = 1\right\}$.

Show that an open subset of a locally compact, Hausdorff space, with its subspace topology, is locally compact.

Show that a closed subset of a locally compact space, with its subspace topology, is locally compact.

Show that a locally compact, Hausdorff space $X$ is compact if and only if the set consisting of the point at infinity is an open subset of the Alexandroff one-point compactification $X^{*}$ of $X$.

Let $X$ be a locally compact, Hausdorff space. Show that the Alexandroff one-point compactification $X^{*}$ is separable if and only if $X$ is separable.

Consider the topological space $X$ consisting of the set of real numbers with the topology that has complements of countable sets as a base. Show that $X$ is not locally compact.

Provide a proof of Proposition 3 by applying Urysohn's Lemma to the Alexandroff one-point compactification of $X$.

Proposition 3: Let $K$ be a compact subset of a locally compact, Hausdorff space $X$ and $O$ a neighborhood of $K$. Then there is a function $f \in C_{c}(X)$ for which $f = 1$ on $K$, $f = 0$ on $X \sim O$ and $0 \leq f \leq 1$ on $X$.

Let $f$ continuously map the locally compact, Hausdorff space $X$ onto the topological space $Y$. Is $Y$ necessarily locally compact?

Let $X$ be a topological space and $f$ a continuous function of $X$ that has compact support. Define $K = \left\{ x \in X \mid f(x) = 1\right\}$. Show that $K$ is a compact $G_{\delta }$ set.

Let $\mathcal{O}$ be an open subset of a compact, Hausdorff space $X$. Show that the mapping of $X$ to the Alexandroff one-point compactification of $\mathcal{O}$ that is the identity on $\mathcal{O}$ and takes each point in $X \sim \mathcal{O}$ into $\omega$ is continuous.

Let $X$ and $Y$ be locally compact, Hausdorff spaces, and $f$ a continuous mapping of $X$ into $Y$. Let $X^{*}$ and $Y^{*}$ be the Alexandroff one-point compactifications of $X$ and $Y$, and $f^{*}$ the mapping of $X^{*}$ into $Y^{*}$ whose restriction to $X$ is $f$ and that takes the point at infinity in $X^{*}$ into the point at infinity in $Y^{*}$. Show that $f^{*}$ is continuous if and only if $f^{-1}(K)$ is compact whenever $K \subseteq Y$ is compact. A mapping $f$ with this property is said to be proper.

Let $X$ be a locally compact, Hausdorff space. Show that a subset $F$ of $X$ is closed if and only if $F \cap K$ is closed for each compact subset $K$ of $X$. Moreover, show that the same equivalence holds if instead of being locally compact the space $X$ is first countable.

Let $\mathcal{F}$ be a family of real-valued continuous functions on a locally compact, Hausdorff space $X$ which has the following properties:

Let $K$ be a compact $G_{\delta }$ subset of a locally compact, Hausdorff space $X$. Show that there is a decreasing sequence of continuous non-negative real-valued functions on $X$ that converges pointwise on $X$ to the characteristic function of $K$.

The Baire Category Theorem asserts that in a complete metric space the intersection of a countable collection of open dense sets is dense. At the heart of its proof lies the Cantor Intersection Theorem. Show that the Fréchet Intersection Theorem is a sufficiently strong substitute for the Cantor Intersection Theorem to provide a proof of the following assertion, by first proving it in the case in which $X$ is compact: Let $X$ be a locally compact, Hausdorff space.

Use the preceding problem to prove the following: Let $X$ be a locally compact, Hausdorff space. If $\mathcal{O}$ is an open subset of $X$ that is contained in a countable union $\bigcup_{n = 1}^{\infty } F_{n}$ of closed subsets of $X$, then the union of their interiors, $\bigcup_{n = 1}^{\infty }$ int $F_{n}$, is an open dense subset of $\mathcal{O}$.

For a map $f: X \rightarrow Y$ and a collection $\mathcal{C}$ of subsets of $Y$ we define $f^{*} \mathcal{C}$ to be the collection of subsets of $X$ given by

Show that if $\mathcal{A}$ is the $\sigma$-algebra generated by $\mathcal{C}$, then $f^{*} \mathcal{A}$ is the $\sigma$-algebra generated by $f^{*} \mathcal{C}$.

For a map $f: X \rightarrow Y$ and a collection $\mathcal{C}$ of subsets of $X$, let $\mathcal{A}$ be the $\sigma$-algebra generated by $\mathcal{C}$. If $f^{-1}[f[C]] = C$ for each $C \in \mathcal{C}$, show that $f^{-1}[f[A]] = A$ for each $A \in \mathcal{A}$.

Show that Euclidean space $\mathbb {R}^{n}$ is locally compact.

Show that $\ell^{p}$, for $1 \leq p \leq \infty$, fails to be locally compact.

Show that $C([0,1])$, with the topology induced by the maximum norm, is not locally compact.

Let $\rho : X \times X \rightarrow \mathbb {R}$ be a metric on a set $X$. For $A \subseteq X$, consider the distance function

Show that property (ii) in the statement of Theorem 7 is equivalent to the assertion that two disjoint closed subsets of $X$, one of which is compact, may be separated by disjoint neighborhoods.

Theorem 7: Let $(X, \mathcal{T})$ be a Hausdorff space. Then the following four properties are equivalent: (i) $(X, \mathcal{T})$ is locally compact; (ii) if $O$ is a neighborhood of a compact subset $K$ of $X$, then there is a neighborhood $U$ of $K$ that has compact closure contained in $O$; (iii) if $O$ is a neighborhood of a compact subset $K$ of $X$, then the constant function on $K$ that takes the value $1$ may be extended to a function $f$ in $C_{c}(X)$ for which $0 \leq f \leq 1$ on $X$ and $f$ vanishes outside of $O$; (iv) for $K$ a compact subset of $X$ and $\mathcal{F}$ a finite open cover of $K$, there is a partition of unity subordinate to $\mathcal{F}$ consisting of functions of compact support.

Let ($X, \mathcal{T}$) be a Hausdorff topological space. Show that $\mathcal{T}$ is a semi-ring if and only if $\mathcal{T}$ is the discrete topology.

Let $(X, \mathcal{T})$ be a topological space and $\mu : \mathcal{T} \rightarrow [0, \infty ]$ a premeasure. Assume that if $\mathcal{O}$ is open and $\mu (\mathcal{O}) < \infty$, then $\mu (\operatorname {bd} \mathcal{O}) = 0$. Show that every open set is $\mu^{*}$-measurable.

Show that the restriction of Lebesgue measure on the real line to the Borel $\sigma$-algebra is a Radon measure.

Show that the restriction of Lebesgue measure on the Euclidean space $\mathbb {R}^{n}$ to the Borel $\sigma$-algebra is a Radon measure.

Show that a Dirac delta measure on a topological space is a Radon measure.

Let $X$ be an uncountable set with the discrete topology and $\left\{ x_{k}\right\}_{1 \leq k < \infty }$ a countable subset of $X$. For $E \subseteq X$, define

Show that $2^{X} = \mathcal{B}(X)$ and $\mu : \mathcal{B}(X) \rightarrow$ is a Radon measure.

Show that the sum of two Radon measures also is Radon.

Let $\mu$ and $\nu$ be Borel measures on $\mathcal{B}(X)$, where $X$ is a compact topological space, and suppose that $\mu$ is absolutely continuous with respect to $\nu$. If $\nu$ is Radon show that $\mu$ also is Radon.

Let $(X, \mathcal{T})$ be a locally compact, Hausdorff space and $\mu : \mathcal{T} \rightarrow [0, \infty ]$ a premeasure for which the restriction to $\mathcal{B}(X)$ of $\mu^{*}$ is a Radon measure. Show that $\mu$ is a Radon premeasure.

Let $X$ be a locally compact, Hausdorff space and $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a Radon measure. Show that any Borel set $E$ of finite measure is inner regular in the sense that

Conclude that if $\mu$ is $\sigma$-finite, then every Borel set is inner regular.

Let $X$ be a topological space, $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a $\sigma$-finite Radon measure, and $E \subseteq X$ a Borel set. Show that there is a $G_{\delta }$ subset $A$ of $X$ and an $F_{\sigma }$ subset $B$ of $X$ for which

For a metric space $X$, show that $\mathcal{B}(X)$ is the smallest $\sigma$-algebra with respect to which all of the continuous real-valued functions on $X$ are measurable.

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

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

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

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

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 $Y$.

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

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 $f \in \mathcal{A}$.

Let $X$ be a separable, compact, Hausdorff space. Show that every closed set is a $G_{\delta }$ set.

Let $X$ be a Hausdorff space and $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ a $\sigma$-finite Borel measure. Show that $\mu$ is Radon if and only if it is regular.

Show that a Hausdorff space $X$ is both locally compact and $\sigma$-compact if and only if there is an ascending countable collection $\left\{ \mathcal{O}_{k}\right\}_{k = 1}^{\infty }$ of open subsets of $X$ that covers $X$ and for each $k$,

Let $x_{0}$ be a point in the locally compact, Hausdorff space $X$. Is the Dirac delta measure concentrated at $x_{0}, \delta_{x_{0}}$, a regular Baire measure?

Let $X$ be an uncountable set with the discrete topology and $X^{*}$ its Alexandroff compactification with $x^{*}$ the point at infinity. Show that the singleton set $\left\{ x^{*}\right\}$ is a Borel set that is not a Baire set.

Let $X$ be a locally compact, Hausdorff space. Show that a Borel measure $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$ is Radon if and only if every Borel set is measurable with respect to the Carathéodory measure induced by the premeasure $\mu : \mathcal{B}(X) \rightarrow [0, \infty ]$.

Prove Lemmas 23, 24, and 25.

Let $X$ be a compact, Hausdorff space and $f_{1}, \ldots , f_{n}$ continuous real-valued functions on $X$. Let $\nu$ be a signed Radon measure on $X$ with $\left|\nu \right|(X) \leq 1$ and let $c_{i} = \int_{X} f_{i} d \nu$, for $1 \leq i \leq n$.