Math Problems Vault

Problem 2B.1

Show that $\mathcal{S} = \left\{ \bigcup_{n \in K}(n, n+1]: K \subseteq \mathbf{Z}\right\}$ is a $\sigma$ -algebra on $\mathbb {R}$ .

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Problem 2B.2

Verify both bullet points in Example 2.28.

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Remark.

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Example 2.28 (smallest $\sigma$ -algebra):

Suppose $X$ is a set and $\mathcal{A}$ is the set of subsets of $X$ that consist of exactly one element: $\mathcal{A} = \left\{ \left\{ x\right\} : x \in X\right\}$ . Then the smallest $\sigma$ -algebra on $X$ containing $\mathcal{A}$ is the set of all subsets $E$ of $X$ such that $E$ is countable or $X \backslash E$ is countable, as you should verify.
Suppose $\mathcal{A} = \left\{ (0,1),(0, \infty )\right\}$ . Then the smallest $\sigma$ -algebra on $\mathbb {R}$ containing $\mathcal{A}$ is $\left\{ \varnothing ,(0,1),(0, \infty ),(-\infty , 0] \cup [1, \infty ),(-\infty , 0],[1, \infty ),(-\infty , 1), \mathbb {R}\right\}$ , as you should verify.

Problem 2B.3

Suppose $\mathcal{S}$ is the smallest $\sigma$ -algebra on $\mathbb {R}$ containing $\left\{ (r, s]: r, s \in \mathbf{Q}\right\}$ . Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$ .

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Problem 2B.4

Suppose $\mathcal{S}$ is the smallest $\sigma$ -algebra on $\mathbb {R}$ containing $\left\{ (r, n]: r \in \mathbf{Q}, n \in \mathbf{Z}\right\}$ . Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$ .

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Problem 2B.5

Suppose $\mathcal{S}$ is the smallest $\sigma$ -algebra on $\mathbb {R}$ containing $\left\{ (r, r+1): r \in \mathbf{Q}\right\}$ . Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$ .

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Problem 2B.6

Suppose $\mathcal{S}$ is the smallest $\sigma$ -algebra on $\mathbb {R}$ containing $\left\{ [r, \infty ): r \in \mathbf{Q}\right\}$ . Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$ .

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Problem 2B.7

Prove that the collection of Borel subsets of $\mathbb {R}$ is translation invariant. More precisely, prove that if $B \subseteq \mathbb {R}$ is a Borel set and $t \in \mathbb {R}$ , then $t+B$ is a Borel set.

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Problem 2B.8

Prove that the collection of Borel subsets of $\mathbb {R}$ is dilation invariant. More precisely, prove that if $B \subseteq \mathbb {R}$ is a Borel set and $t \in \mathbb {R}$ , then $t B$ (which is defined to be $\left\{ t b: b \in B\right\}$ ) is a Borel set.

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Problem 2B.9

Give an example of a measurable space $(X, \mathcal{S})$ and a function $f: X \rightarrow \mathbb {R}$ such that $\left|f\right|$ is $\mathcal{S}$ -measurable but $f$ is not $\mathcal{S}$ -measurable.

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Problem 2B.10

Show that the set of real numbers that have a decimal expansion with the digit 5 appearing infinitely often is a Borel set.

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Problem 2B.11

(a)(b)

Suppose $\mathcal{T}$ is a $\sigma$ -algebra on a set $Y$ and $X \in \mathcal{T}$ . Let $\mathcal{S} = \left\{ E \in \mathcal{T}: E \subseteq X\right\}$ .

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(a)

Show that $\mathcal{S} = \left\{ F \cap X: F \in \mathcal{T}\right\}$ .

(b)

Show that $\mathcal{S}$ is a $\sigma$ -algebra on $X$ .

Problem 2B.12

(a)(b)(c)

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a function.

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(a)

For $k \in \mathbb {N}$ , let

G_{k} = \left\{ a \in \mathbb {R}: \text{ there exists } \delta > 0 \text{ such that }\left|f(b)-f(c)\right| < \frac{1}{k} \text{ for all } b, c \in (a-\delta , a+\delta )\right\}

Prove that $G_{k}$ is an open subset of $\mathbb {R}$ for each $k \in \mathbb {N}$ .

(b)

Prove that the set of points at which $f$ is continuous equals $\bigcap_{k = 1}^{\infty } G_{k}$ .

(c)

Conclude that the set of points at which $f$ is continuous is a Borel set.

Problem 2B.13

Suppose $(X, \mathcal{S})$ is a measurable space, $E_{1}, \ldots , E_{n}$ are disjoint subsets of $X$ , and $c_{1}, \ldots , c_{n}$ are distinct nonzero real numbers. Prove that $c_{1} \chi_{E_{1}}+\cdots +c_{n} \chi_{E_{n}}$ is an $\mathcal{S}$ -measurable function if and only if $E_{1}, \ldots , E_{n} \in \mathcal{S}$ .

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Problem 2B.14

(a)(b)

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(a)

Suppose $f_{1}, f_{2}, \ldots$ is a sequence of functions from a set $X$ to $\mathbb {R}$ . Explain why

\begin{aligned} & \left\{ x \in X: \text{ the sequence } f_{1}(x), f_{2}(x), \ldots \text{ has a limit in } \mathbb {R}\right\} \\ & = \bigcap _{n = 1}^{\infty } \bigcup _{j = 1}^{\infty } \bigcap _{k = j}^{\infty }\left(f_{j}-f_{k}\right)^{-1}\left(\left(-\frac{1}{n}, \frac{1}{n}\right)\right) \end{aligned}

(b)

Suppose $(X, \mathcal{S})$ is a measurable space and $f_{1}, f_{2}, \ldots$ is a sequence of $\mathcal{S}$ measurable functions from $X$ to $\mathbb {R}$ . Prove that

\left\{ x \in X: \text{ the sequence } f_{1}(x), f_{2}(x), \ldots \text{ has a limit in } \mathbb {R}\right\}

is an $\mathcal{S}$ -measurable subset of $X$ .

Problem 2B.15

(a)(b)

Suppose $X$ is a set and $E_{1}, E_{2}, \ldots$ is a disjoint sequence of subsets of $X$ such that $\bigcup_{k = 1}^{\infty } E_{k} = X$ . Let $\mathcal{S} = \left\{ \bigcup_{k \in K} E_{k}: K \subseteq \mathbb {N}\right\}$ .

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(a)

Show that $\mathcal{S}$ is a $\sigma$ -algebra on $X$ .

(b)

Prove that a function from $X$ to $\mathbb {R}$ is $\mathcal{S}$ -measurable if and only if the function is constant on $E_{k}$ for every $k \in \mathbb {N}$ .

Problem 2B.16

(a)(b)

Suppose $\mathcal{S}$ is a $\sigma$ -algebra on a set $X$ and $A \subseteq X$ . Let

\mathcal{S}_{A} = \left\{ E \in \mathcal{S}: A \subseteq E \text{ or } A \cap E = \varnothing \right\} .

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(a)

Prove that $\mathcal{S}_{A}$ is a $\sigma$ -algebra on $X$ .

(b)

Suppose $f: X \rightarrow \mathbb {R}$ is a function. Prove that $f$ is measurable with respect to $\mathcal{S}_{A}$ if and only if $f$ is measurable with respect to $\mathcal{S}$ and $f$ is constant on $A$ .

Problem 2B.17

Suppose $X$ is a Borel subset of $\mathbb {R}$ and $f: X \rightarrow \mathbb {R}$ is a function such that $\left\{ x \in X: f\text{ is not continuous at }x\right\}$ is a countable set. Prove $f$ is a Borel measurable function.

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Problem 2B.18

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is differentiable at every element of $\mathbb {R}$ . Prove that $f^{\prime }$ is a Borel measurable function from $\mathbb {R}$ to $\mathbb {R}$ .

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Problem 2B.19

Suppose $X$ is a nonempty set and $\mathcal{S}$ is the $\sigma$ -algebra on $X$ consisting of all subsets of $X$ that are either countable or have a countable complement in $X$ . Give a characterization of the $\mathcal{S}$ -measurable real-valued functions on $X$ .

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Problem 2B.20

Suppose $(X, \mathcal{S})$ is a measurable space and $f, g: X \rightarrow \mathbb {R}$ are $\mathcal{S}$ -measurable functions. Prove that if $f(x) > 0$ for all $x \in X$ , then $f^{g}$ (which is the function whose value at $x \in X$ equals $f(x)^{g(x)}$ ) is an $\mathcal{S}$ -measurable function.

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Problem 2B.21

Prove 2.52.

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Remark.

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2.52 (condition for measurable function): Suppose $(X, \mathcal{S})$ is a measurable space and $f: X \rightarrow [-\infty , \infty ]$ is a function such that $f^{-1}(a, \infty ] \in \mathcal{S}$ for all $a \in \mathbb {R}$ . Then $f$ is an $\mathcal{S}$ -measurable function.

Problem 2B.22

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is an increasing function. Prove that $f$ is continuous at every element of $B$ except for a countable subset of $B$ .

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Problem 2B.23

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a strictly increasing function. Prove that the inverse function $f^{-1}: f(\mathbb {R}) \rightarrow \mathbb {R}$ is a continuous function.

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Remark.

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Note that this exercise does not have as a hypothesis that $f$ is continuous.

Problem 2B.24

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a strictly increasing function and $B \subseteq \mathbb {R}$ is a Borel set. Prove that $f(B)$ is a Borel set.

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Problem 2B.25

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is an increasing function. Prove that there exists a sequence $f_{1}, f_{2}, \ldots$ of strictly increasing functions from $B$ to $\mathbb {R}$ such that

f(x) = \lim _{k \rightarrow \infty } f_{k}(x)

for every $x \in B$ .

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Problem 2B.26

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is a bounded increasing function. Prove that there exists an increasing function $g: \mathbb {R} \rightarrow \mathbb {R}$ such that $g(x) = f(x)$ for all $x \in B$ .

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Problem 2B.27

Prove or give a counterexample: If $(X, \mathcal{S})$ is a measurable space and

f: X \rightarrow [-\infty , \infty ]

is a function such that $f^{-1}((a, \infty )) \in \mathcal{S}$ for every $a \in \mathbb {R}$ , then $f$ is an $\mathcal{S}$ -measurable function.

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Problem 2B.28

Suppose $f: B \rightarrow \mathbb {R}$ is a Borel measurable function. Define $g: \mathbb {R} \rightarrow \mathbb {R}$ by

g(x) = \begin{cases} f(x) & \text{ if } x \in B \\ 0 & \text{ if } x \in \mathbb {R} \backslash B\end{cases}

Prove that $g$ is a Borel measurable function.

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Problem 2B.29

Give an example of a measurable space $(X, \mathcal{S})$ and a family $\left\{ f_{t}\right\}_{t \in \mathbb {R}}$ such that each $f_{t}$ is an $\mathcal{S}$ -measurable function from $X$ to $[0,1]$ , but the function $f: X \rightarrow [0,1]$ defined by

f(x) = \sup \left\{ f_{t}(x): t \in \mathbb {R}\right\}

is not $\mathcal{S}$ -measurable.

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Remark.

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Compare this exercise to 2.53, where the index set is $\mathbb {N}$ rather than $\mathbb {R}$ .

Problem 2B.30

Show that

\lim _{j \rightarrow \infty }\left(\lim _{k \rightarrow \infty }(\cos (j!\pi x))^{2 k}\right) = \begin{cases} 1 & \text{ if } x \text{ is rational } \\ 0 & \text{ if } x \text{ is irrational }\end{cases}

for every $x \in \mathbb {R}$ .

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Remark.

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This example is due to Henri Lebesgue.