Math Problems Vault

Problem 5C.1

Show that a set $G \subseteq \mathbb {R}^{n}$ is open in $\mathbb {R}^{n}$ if and only if for each $\left(b_{1}, \ldots , b_{n}\right) \in G$ , there exists $r > 0$ such that

\left\{ \left(a_{1}, \ldots , a_{n}\right) \in \mathbb {R}^{n}: \sqrt{\left(a_{1}-b_{1}\right)^{2}+\cdots +\left(a_{n}-b_{n}\right)^{2}} < r\right\} \subseteq G

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Problem 5C.2

Show that there exists a set $E \subseteq \mathbb {R}^{2}$ (thinking of $\mathbb {R}^{2}$ as equal to $\mathbb {R} \times \mathbb {R}$ ) such that the cross sections $[E]_{a}$ and $[E]^{a}$ are open subsets of $\mathbb {R}$ for every $a \in \mathbb {R}$ , but $E \notin \mathcal{B}_{2}$ .

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Problem 5C.3

Suppose $(X, \mathcal{S}),(Y, \mathcal{T})$ , and $(Z, \mathcal{U})$ are measurable spaces. We can define $\mathcal{S} \otimes \mathcal{T} \otimes \mathcal{U}$ to be the smallest $\sigma$ -algebra on $X \times Y \times Z$ that contains

\left\{ A \times B \times C: A \in \mathcal{S}, B \in \mathcal{T}, C \in \mathcal{U}\right\}

Prove that if we make the obvious identifications of the products $(X \times Y) \times Z$ and $X \times (Y \times Z)$ with $X \times Y \times Z$ , then

\mathcal{S} \otimes \mathcal{T} \otimes \mathcal{U} = (\mathcal{S} \otimes \mathcal{T}) \otimes \mathcal{U} = \mathcal{S} \otimes (\mathcal{T} \otimes \mathcal{U})

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Problem 5C.4

Show that Lebesgue measure on $\mathbb {R}^{n}$ is translation invariant. More precisely, show that if $E \in \mathcal{B}_{n}$ and $a \in \mathbb {R}^{n}$ , then $a+E \in \mathcal{B}_{n}$ and $\lambda_{n}(a+E) = \lambda_{n}(E)$ , where

a+E = \left\{ a+x: x \in E\right\}

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Problem 5C.5

(a)(b)

Suppose $f: \mathbb {R}^{n} \rightarrow \mathbb {R}$ is $\mathcal{B}_{n}$ -measurable and $t \in \mathbb {R} \backslash \left\{ 0\right\}$ . Define $f_{t}: \mathbb {R}^{n} \rightarrow \mathbb {R}$ by $f_{t}(x) = f(t x)$ .

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(a)

Prove that $f_{t}$ is $\mathcal{B}_{n}$ -measurable.

(b)

Prove that if $\int_{\mathbb {R}^{n}} f d \lambda_{n}$ is defined, then

\int _{\mathbb {R}^{n}} f_{t} d \lambda _{n} = \frac{1}{\left|t\right|^{n}} \int _{\mathbb {R}^{n}} f d \lambda _{n}

Problem 5C.6

Suppose $\lambda$ denotes Lebesgue measure on $(\mathbb {R}, \mathcal{L})$ , where $\mathcal{L}$ is the $\sigma$ -algebra of Lebesgue measurable subsets of $\mathbb {R}$ . Show that there exist subsets $E$ and $F$ of $\mathbb {R}^{2}$ such that

$F \in \mathcal{L} \otimes \mathcal{L}$ and $(\lambda \times \lambda )(F) = 0$ ;
$E \subseteq F$ but $E \notin \mathcal{L} \otimes \mathcal{L}$ .

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Remark.

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The measure space $(\mathbb {R}, \mathcal{L}, \lambda )$ has the property that every subset of a measurable set with measure 0 is measurable. This exercise asks you to show that the measure space $\left(\mathbb {R}^{2}, \mathcal{L} \otimes \mathcal{L}, \lambda \times \lambda \right)$ does not have this property.

Problem 5C.7

Suppose $m \in \mathbb {N}$ . Verify that the collection of sets $\mathcal{E}_{m}$ that appears in the proof of 5.41 is a monotone class.

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Problem 5C.8

Show that the open unit ball in $\mathbb {R}^{n}$ is an open subset of $\mathbb {R}^{n}$ .

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Problem 5C.9

Suppose $G_{1}$ is a nonempty subset of $\mathbb {R}^{m}$ and $G_{2}$ is a nonempty subset of $\mathbb {R}^{n}$ . Prove that $G_{1} \times G_{2}$ is an open subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ if and only if $G_{1}$ is an open subset of $\mathbb {R}^{m}$ and $G_{2}$ is an open subset of $\mathbb {R}^{n}$ .

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Remark.

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One direction of this result was already proved (see 5.36); both directions are stated here to make the result look prettier and to be comparable to the next exercise, where neither direction has been proved.

Problem 5C.10

Suppose $F_{1}$ is a nonempty subset of $\mathbb {R}^{m}$ and $F_{2}$ is a nonempty subset of $\mathbb {R}^{n}$ . Prove that $F_{1} \times F_{2}$ is a closed subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ if and only if $F_{1}$ is a closed subset of $\mathbb {R}^{m}$ and $F_{2}$ is a closed subset of $\mathbb {R}^{n}$ .

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Problem 5C.11

(a)(b)

Suppose $E$ is a subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ and

A = \left\{ x \in \mathbb {R}^{m}:(x, y) \in E \text{ for some } y \in \mathbb {R}^{n}\right\}

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(a)

Prove that if $E$ is an open subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ , then $A$ is an open subset of $\mathbb {R}^{m}$ .

(b)

Prove or give a counterexample: If $E$ is a closed subset of $\mathbb {R}^{m} \times \mathbb {R}^{n}$ , then $A$ is a closed subset of $\mathbb {R}^{m}$ .

Problem 5C.12

(a)(b)

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(a)

Prove that $\lim_{n \rightarrow \infty } \lambda_{n}\left(\mathbf{B}_{n}\right) = 0$ .

(b)

Find the value of $n$ that maximizes $\lambda_{n}\left(\mathbf{B}_{n}\right)$ .

Problem 5C.13

For readers familiar with the gamma function $\Gamma$ : Prove that

\lambda _{n}\left(\mathbf{B}_{n}\right) = \frac{\pi ^{n / 2}}{\Gamma \left(\frac{n}{2}+1\right)}

for every positive integer $n$ .

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Problem 5C.14

(a)(b)(c)

Define $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ by

f(x, y) = \begin{cases} \frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}} & \text{ if }(x, y) \neq (0,0) \\ 0 & \text{ if }(x, y) = (0,0)\end{cases}

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(a)

Prove that $D_{1}\left(D_{2} f\right)$ and $D_{2}\left(D_{1} f\right)$ exist everywhere on $\mathbb {R}^{2}$ .

(b)

Show that $\left(D_{1}\left(D_{2} f\right)\right)(0,0) \neq \left(D_{2}\left(D_{1} f\right)\right)(0,0)$ .

(c)

Explain why (b) does not violate 5.48.

Remark.

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5.48 (equality of mixed partial derivatives): Suppose $G$ is an open subset of $\mathbb {R}^{2}$ and $f: G \rightarrow \mathbb {R}$ is a function such that $D_{1} f$ , $D_{2} f$ , $D_{1}\left(D_{2} f\right)$ , and $D_{2}\left(D_{1} f\right)$ all exist and are continuous functions on $G$ . Then $D_{1}\left(D_{2} f\right) = D_{2}\left(D_{1} f\right)$ on $G$ .