Particular Measures

Show that the Cauchy-Schwarz Inequality is an equality if and only if the points are linearly dependent.

Verify the following generalization of the Pythagorean Identity called the Parallelogram Law:

For $1 \leq p \leq \infty$, define the $\left\| \cdot \right\|_{p}$ norm on $\mathbb {R}^{n}$ and prove the Hölder and Minkowski Inequalities. Show that for all $1 \leq \alpha , \beta \leq \infty$, there is a constant $c = c(\alpha , \beta )$ for which

Show that, in Urysohn's Lemma, if $A$ fails to be compact, there may not be a compactly supported, separating function $f$.

Show that Lebesgue outer-measure on $\mathbb {R}$ is changed if, following Jordan, countable covers are restricted to finite covers.

For the cube $Q \subseteq \mathbb {R}^{3}$ with each side $[0,1]$, let $Q$ be the disjoint union of $k$ intervals. Prove additivity of volume for this partition without using Lemma 1.

Let $A$ and $B$ be of $\mathbb {R}^{n}$ for which there is a $c > 0$ such that $\left\| u-v\right\| \geq c$ for all $u \in A, v \in B$. Show that

Prove that $\mathcal{L}^{n}$ is the smallest $\sigma$-algebra that contains all Borel sets and all sets of Lebesgue measure zero.

If $E \in \mathcal{L}^{n}$ and $\epsilon > 0$, is there a compact set $F$ such that $F \subseteq E$ and $\mu_{n}(E \sim F) < \epsilon$ ?

Let $E$ be any set of real numbers. Show that $E \times \mathbb {Q}$ is a Lebesgue measurable subset of $\mathbb {R}^{2}$, regardless of the Lebesgue measurability of $E \subseteq \mathbb {R}$.

Show that if the mapping $\mathcal{N}: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is continuous and maps sets of Lebesgue measure zero to sets of Lebesgue measure zero, then it maps Lebesgue measurable sets to Lebesgue measurable sets.

Consider the triangle $\Delta = \left\{ (x, y) \in \mathbb {R}^{2} \mid 0 \leq x \leq a, 0 \leq y \leq [b / a] x\right\}$. By covering $\Delta$ with finite collections of rectangles and using the continuity of measure, determine the Lebesgue measure of $\Delta$.

Let $[a, b]$ be a closed, bounded interval. Suppose that $f:[a, b] \rightarrow \mathbb {R}$ is bounded and Lebesgue measurable. Show that the graph of $f$ has measure zero with respect to Lebesgue measure on the plane. Generalize this to bounded, real-valued measurable functions of several real variables.

Show that a continuously differentiable mapping $\mathcal{N}: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ may not be Lipschitz, but its restriction to each closed, bounded set is.

For a linear operator, compare the estimate (6) with the actual measure.

Let $R \subseteq \mathbb {R}^{n}$ be a bounded rectangle and $\epsilon > 0$. Show that there is a finite cover of $R$ by cubes, the sum of whose volumes is less than $\operatorname {vol}(R)+\epsilon$. Conclude from this that Lebesgue outer-measure on $\mathbb {R}^{n}$ is unaltered if coverings by intervals are restricted to coverings by cubes.

Show that each open subset of $\mathbb {R}^{n}$ is an $F_{\sigma }$ set.

Show that the open, bounded subset of the plane $(0,1) \times (0,1)$ is not the union of a countable, disjoint collection of open balls in $\mathbb {R}^{2}$. Does this contradict the Vitali Partition Theorem?

Show that for $n = 1$ the constant $C$ in (6) cannot be improved.

Show that the Cauchy-Schwarz Inequality is an equality if and only if $x$ and $y$ are linearly dependent. Use this to show that $\left\| L\right\|$ is the smallest Lipchitz constant for a linear operator $L: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$.

A pair of points in $\mathbb {R}^{n}$ is said to be orthogonal provided that their inner-product is zero. A set $\left\{ v_{1}, \ldots v_{n}\right\} \subseteq \mathbb {R}^{n}$ is said to be an orthonormal basis for $\mathbb {R}^{n}$ provided that each pair of points in the set is orthogonal, and each point has norm 1. For a linear operator $L: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$, prove the equivalence of the following assertions:

Show that, in $\mathbb {R}^{3}$, Corollary 22 reduces to the triple-product formula for volume.

For $\left\{ z_{1}, \ldots , z_{n}\right\} \subseteq \mathbb {R}^{n}$, let $P(z_{1}, \ldots , z_{n})$ denote the parallelepiped consisting of the points $\sum_{i=1}^{n} x_{i} \cdot z_{i}$ with coefficients $0 \leq x_{i} \leq 1$, and let $\det (z_{1}, \ldots , z_{n})$ be the determinant of the $n \times n$ matrix whose $i$-th column is $z_{i}$. Corollary 22: For $\left\{ z_{1}, \ldots , z_{n}\right\} \subseteq \mathbb {R}^{n}$, $\mu_{n}(P(z_{1}, \ldots , z_{n})) = \left|\det (z_{1}, \ldots , z_{n})\right|$.

Prove that for any open cover of a closed, bounded subset of $\mathbb {R}^{n}$ there is an $\alpha > 0$ (called the Lebesgue number of the cover) such that every ball centered in $K$ and of radius $\alpha$ is contained in one of the covering sets.

The collection of invertible linear operators on $\mathbb {R}^{n}$ is called the general linear group and denoted by $G L(n)$; the collection of orthogonal operators is denoted by $\mathcal{O}(n)$ and called the orthogonal group. Show that with the group operation of composition, $G L(n)$ is a group that has $\mathcal{O}(n)$ as a subgroup.

Prove the inequality (18).

For a finite measure $\mu$ on $\mathcal{B}(\mathbb {R})$, define $g: \mathbb {R} \rightarrow \mathbb {R}$ by setting $g(x) = \mu (-\infty , x]$. Show that each bounded, increasing function $g: \mathbb {R} \rightarrow \mathbb {R}$ that is continuous on the right and $\lim_{x \rightarrow -\infty } g(x) = 0$ is the cumulative distribution function of a unique finite Borel measure on $\mathcal{B}(\mathbb {R})$.

Show that every closed subset of $\mathbb {R}^{n}$ is a $G_{\delta }$ set, and every open subset of $\mathbb {R}^{n}$ is an $F_{\sigma }$ set.

Show that Theorem 24 may not hold if $\mu \left(\mathbb {R}^{n}\right) = \infty$.

Theorem 24: Every finite Borel measure $\mu : \mathscr {B}\left(\mathbb {R}^{n}\right) \rightarrow [0, \infty )$ is regular, that is, for every $\epsilon > 0$ and every set $E \in \mathscr {B}\left(\mathbb {R}^{n}\right)$, there is a compact set $K$ and open set $O$ for which $K \subseteq E \subseteq O$, $\mu (E \sim K) < \epsilon$ and $\mu (O \sim E) < \epsilon$.

Show that the regularity of Lebesgue measure on $\mathbb {R}^{n}$ is a consequence of Theorem 24.

Theorem 24: Every finite Borel measure $\mu : \mathscr {B}\left(\mathbb {R}^{n}\right) \rightarrow [0, \infty )$ is regular, that is, for every $\epsilon > 0$ and every set $E \in \mathscr {B}\left(\mathbb {R}^{n}\right)$, there is a compact set $K$ and open set $O$ for which $K \subseteq E \subseteq O$, $\mu (E \sim K) < \epsilon$ and $\mu (O \sim E) < \epsilon$.

Directly from the definition of Lebesgue outer-measure, show that if $A$ and $B$ are subsets of $\mathbb {R}^{n}$ and there is a $\delta > 0$ for which $\left\| u-v\right\| \geq \delta$ for all $u \in A$ and $v \in B$, then

Show that in the definition of Hausdorff measure one can take the coverings to be by open sets or by closed sets.

Show that the set-function outer Hausdorff measure $H_{\alpha }^{*}$ is countably monotone.

In the plane $\mathbb {R}^{2}$ show that a bounded set may be enclosed in a ball of the same diameter. Use this to show that for a bounded subset $A$ of $\mathbb {R}^{2}, H_{2}(A) \geq 4 / \pi \cdot \mu_{2}(A)$, where $\mu_{2}$ is Lebesgue measure on $\mathbb {R}^{2}$.

Let ($X, \rho$) be a metric space and $\alpha > 0$. For $E \subseteq X$, define

where $\left\{ A_{k}\right\}_{k = 1}^{\infty }$ is a countable collection of subsets of $X$ that covers $E$ : there is no restriction regarding the size of the diameters of the sets in the cover. Compare the set-functions $H_{\alpha }^{\prime }$ and $H_{\alpha }$.

Show that each Hausdorff measure $H_{\alpha }$ on Euclidean space $\mathbb {R}^{n}$ is invariant with respect to constant perturbations of orthogonal operators.

Give a direct proof to show that if $I$ is a nontrivial interval in $\mathbb {R}^{n}$, then $H_{n}(I) > 0$.

Show that in any metric space, $H_{0}$ is counting measure.

Let $[a, b]$ be a closed, bounded interval on real numbers and consider the subset of the plane defined by $\mathbb {R} = \left\{ (x, y) \in \mathbb {R}^{2} \mid a \leq x \leq b, y = 0\right\}$. Show that $H_{2}(R) = 0$. Then show that $H_{1}(R) = b-a$. Conclude that the Hausdorff dimension of $\mathbb {R}$ is 1.

Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous, bounded function on the closed, bounded interval $[a, b]$ that has a continuous, bounded derivative on the open interval ($a, b$). Consider the graph $G$ of $f$ as a subset of the plane. Show that $H_{1}(G) = \int_{a}^{b} \sqrt{1+\left|f^{\prime }(x)\right|^{2}} d x$.

Let $J$ be an interval in $\mathbb {R}^{n}$, each of whose sides has length 1. Define $\gamma_{n} = H_{n}(J)$. Show that if $I$ is any bounded interval in $\mathbb {R}^{n}$, then $H_{n}(I) = \gamma_{n} \cdot \mu_{n}(I)$. From this conclude, using the uniqueness assertion of the Carathéodory-Hahn Theorem, that $H_{n} = \gamma_{n} \cdot \mu_{n}$ on the Borel subsets of $\mathbb {R}^{n}$.

Show that the difference of two measures, one of which is finite, is a signed measure.

Show that the Hahn Decomposition may not be unique.

Let $\mu$ be a measure and $\mu_{1}$ and $\mu_{2}$ be mutually singular measures on a $\sigma$-algebra $\mathcal{M}$ for which $\mu = \mu_{1}-\mu_{2}$. Show that $\mu_{2} = 0$. Use this to establish the uniqueness of the Jordan Decomposition.

Show that if $\nu_{1}$ and $\nu_{2}$ are any two finite signed measures, then for any $\alpha$ and $\beta$, so is $\alpha \nu_{1}+\beta \nu_{2}$. Show that

where $\nu \leq \mu$ means $\nu (E) \leq \mu (E)$ for all measurable sets $E$.

Let $\mu$ and $\nu$ be finite signed measures. Define $\mu \wedge \nu = \frac{1}{2}(\mu +\nu -\left|\mu -\nu \right|)$ and $\mu \vee \nu = \mu +\nu -\mu \wedge \nu$.