Lebesgue Measurability and Measure of Images of Mappings

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.