Math Problems Vault

Problem 1.1

Consider the measurable space $(\Omega , \mathcal{A})$ and let $A_{n} \in \mathcal{A}, n = 1,2, \ldots$ Then recall that

\liminf _{n \rightarrow \infty } A_{n} = \underline{\lim }_{n \rightarrow \infty } A_{n} = \bigcup _{n = 1}^{\infty } \bigcap _{j = n}^{\infty } A_{j} \text{ and } \limsup _{n \rightarrow \infty } A_{n} = \varlimsup _{n \rightarrow \infty } A_{n} = \bigcap _{n = 1}^{\infty } \bigcup _{j = n}^{\infty } A_{j}

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(i)

Show that $\underline{\lim }_{n \rightarrow \infty } A_{n} \subseteq \overline{\lim }_{n \rightarrow \infty } A_{n}$ . (If also $\overline{\lim }_{n \rightarrow \infty } A_{n} \subseteq \underline{\lim }_{n \rightarrow \infty } A_{n}$ , so that $\underline{\lim }_{n \rightarrow \infty } A_{n} = \varlimsup_{n \rightarrow \infty } A_{n}$ , then this set is denoted by $\lim_{n \rightarrow \infty } A_{n}$ and is called the limit of the sequence $\left\{ A_{n}\right\} , n \geq 1$ .)

(ii)

Show that $\left(\underline{\lim }_{n \rightarrow \infty } A_{n}\right)^{c} = \varlimsup_{n \rightarrow \infty } A_{n}^{c}$ , $\left(\varlimsup_{n \rightarrow \infty } A_{n}\right)^{c} = \underline{\lim }_{n \rightarrow \infty } A_{n}^{c}$ . Conclude that if $\lim_{n \rightarrow \infty } A_{n} = A$ , then $\lim_{n \rightarrow \infty } A_{n}^{c} = A^{c}$ .

(iii)

Show that

\underline{\lim }_{n \rightarrow \infty }\left(A_{n} \cap B_{n}\right) = \left(\underline{\lim }_{n \rightarrow \infty } A_{n}\right) \cap \left(\underline{\lim }_{n \rightarrow \infty } B_{n}\right),

and

\varlimsup _{n \rightarrow \infty }\left(A_{n} \cup B_{n}\right) = \left(\varlimsup _{n \rightarrow \infty } A_{n}\right) \cup \left(\varlimsup _{n \rightarrow \infty } B_{n}\right)

(iv)

Show that

\varlimsup _{n \rightarrow \infty }\left(A_{n} \cap B_{n}\right) \subseteq \left(\varlimsup _{n \rightarrow \infty } A_{n}\right) \cap \left(\varlimsup _{n \rightarrow \infty } B_{n}\right)

and

\underline{\lim }_{n \rightarrow \infty }\left(A_{n} \cup B_{n}\right) \supseteq \left(\underline{\lim }_{n \rightarrow \infty } A_{n}\right) \cup \left(\underline{\lim }_{n \rightarrow \infty } B_{n}\right) .

(v)

By a counterexample, show that the inverse inclusions in part (iv) do not hold, so that $\overline{\lim }_{n \rightarrow \infty }\left(A_{n} \cap B_{n}\right)$ need not be equal to $\left(\overline{\lim }_{n \rightarrow \infty } A_{n}\right) \cap \left(\overline{\lim }_{n \rightarrow \infty } B_{n}\right)$ , and $\underline{\lim }_{n \rightarrow \infty }\left(A_{n} \cup B_{n}\right)$ need not be equal to $\left(\underline{\lim }_{n \rightarrow \infty } A_{n}\right) \cup \left(\underline{\lim }_{n \rightarrow \infty } B_{n}\right)$ .

(vi)

If $\lim_{n \rightarrow \infty } A_{n} = A$ and $\lim_{n \rightarrow \infty } B_{n} = B$ , then show that $\lim_{n \rightarrow \infty }\left(A_{n} \cap B_{n}\right) = A \cap B$ and $\lim_{n \rightarrow \infty }\left(A_{n} \cup B_{n}\right) = A \cup B$ .

(vii)

If $\lim_{n \rightarrow \infty } A_{n} = A$ , then show that for any set $B, \lim_{n \rightarrow \infty }\left(A_{n} \Delta B\right) = A \Delta B$ , where $A_{n} \Delta B$ is the symmetric difference of $A_{n}$ and $B$ .

(viii)

If $A_{2 j-1} = B$ and $A_{2 j} = C, j = 1,2, \ldots$ , determine $\underline{\lim }_{n \rightarrow \infty } A_{n}$ and $\varlimsup_{n \rightarrow \infty } A_{n}$ . Under what condition on $B$ and $C$ does the limit exist, and what is it equal to?

Problem 1.2

(i)(ii)

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(i)

Setting $\underline{A} = \liminf_{n \rightarrow \infty } A_{n}, \bar{A} = \lim \sup_{n \rightarrow \infty } A_{n}$ , and $A = \lim_{n \rightarrow \infty } A_{n}$ if it exists, show that all $\underline{A}, \bar{A}$ , and $A$ are in $\mathcal{A}$ .

(ii)

If $A_{n} \uparrow$ as $n \rightarrow \infty$ , show that $\lim_{n \rightarrow \infty } A_{n}$ exists and is equal to $\cup_{n = 1}^{\infty } A_{n}$ , and if $A_{n} \downarrow$ as $n \rightarrow \infty$ , then $\lim_{n \rightarrow \infty } A_{n}$ exists and is equal to $\cap_{n = 1}^{\infty } A_{n}$ .

Problem 1.3

Recall the following theorem (Theorem 1). Let $I$ be any index set. Then

If $\mathcal{F}_{j}, j \in I$ , are fields, so is $\cap_{j \in I} \mathcal{F}_{j} = \left\{ A \subseteq \Omega ; A \in \mathcal{F}_{j}, j \in I\right\}$ .
If $\mathcal{A}_{j}, j \in I$ , are $\sigma$ -fields, so is $\cap_{j \in I} \mathcal{A}_{j} = \left\{ A \subseteq \Omega ; A \in \mathcal{A}_{j}, j \in I\right\}$ .

Carry out the details of the proof of Theorem 1.

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Problem 1.4

By means of an example, show that $A_{j} \in \mathcal{F}, j \geq 1$ , need not imply that $\bigcup_{j = 1}^{\infty } A_{j} \in \mathcal{F}$ , and similarly for $\bigcap_{j = 1}^{\infty } A_{j}$ .

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Problem 1.5

Let $\mathcal{P} = \left\{ A_{n}, n = 1,2, \ldots \right\}$ be a partition of $\Omega$ where $A_{n} \neq \varnothing , n \geq 1$ , and let $\mathcal{C}$ be the class of all sums of members in $\mathcal{P}$ . Then show that $\mathcal{C}$ is the $\sigma$ -field generated by the class $\mathcal{P}$ .

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Problem 1.6

Let $\mathcal{C}_{0}$ be the class of all intervals in $\mathbb {R}$ , and consider the eight classes $\mathcal{C}_{j}$ , $j = 1, \ldots , 8$ , each of which consists of all intervals in $\mathcal{C}_{0}$ of one type. Then $\mathcal{B} = \sigma \left(\mathcal{C}_{j}\right), j = 1, \ldots , 8$ . Also, if $\mathcal{C}_{j}^{\prime }$ denotes the class we get from $\mathcal{C}_{j}$ by considering intervals with rational endpoints, then $\sigma \left(\mathcal{C}_{j}^{\prime }\right) = \mathcal{B}, j = 1, \ldots , 8$ .

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Problem 1.7

(i)(ii)

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(i)

If $\mathcal{C}$ is the class of all finite sums of intervals in $\mathbb {R}$ (unions of pairwise disjoint intervals) of the form: $(\alpha , \beta ], \alpha , \beta \in \mathbb {R}, \alpha < \beta ;(-\infty , \alpha ]$ , $\alpha \in \mathbb {R} ;(\beta , \infty ), \beta \in \mathbb {R}, \oslash$ , then $\mathcal{C}$ is a field and $\sigma (\mathcal{C}) = \mathcal{B}$ .

(ii)

The same is true if $\mathcal{C}$ is the class of all finite sums of all kinds of intervals in $\mathbb {R}$ .

Problem 1.8

Consider the space $(\Omega , \mathcal{F})$ and for an arbitrary but fixed set $A$ with $\oslash \subset A \subset \Omega$ , define $\mathcal{F}_{A}$ by: $\mathcal{F}_{A} = \left\{ B \subseteq \Omega ; B = A \cap C, C \in \mathcal{F}\right\}$ . Then $\mathcal{F}_{A}$ is a field (of subsets of $A$ ).

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Problem 1.9

Consider the space $(\Omega , \mathcal{A})$ and let $A$ be as in Exercise 8. Define $\mathcal{A}_{A}$ by $\mathcal{A}_{A} = \left\{ B \subseteq \Omega ; B = A \cap C, C \in \mathcal{A}\right\}$ . Then $\mathcal{A}_{A}$ is a $\sigma$ -field (of subsets of $A$ ). Furthermore, $\mathcal{A}_{A} = \sigma \left(\mathcal{F}_{A}\right)$ , where $\mathcal{F}_{A}$ is as in Exercise 8 and $\mathcal{A} = \sigma (\mathcal{F})$ .

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Problem 1.10

Show that, if $\left\{ \mathcal{A}_{n}\right\} , n \geq 1$ , is a nondecreasing sequence of $\sigma$ -fields, then $\bigcup_{n = 1}^{\infty } \mathcal{A}_{n}$ is always a field, but it may fail to be a $\sigma$ -field.

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Problem 1.11

Recall (Definition 4) that a class $\mathcal{C}$ of subsets of $\Omega$ is called a monotone class if, whenever $A_{j} \in \mathcal{C}, j = 1, 2, \ldots$ , and $A_{j} \uparrow$ (i.e., $A_{1} \subseteq A_{2} \subseteq \cdots$ ) or $A_{j} \downarrow$ (i.e., $A_{1} \supseteq A_{2} \supseteq \cdots$ ), then $\lim_{j \rightarrow \infty } A_{j} \stackrel{\text{def}}{=} \cup_{j=1}^{\infty } A_{j} \in \mathcal{C}$ and $\lim_{j \rightarrow \infty } A_{j} \stackrel{\text{def}}{=} \cap_{j=1}^{\infty } A_{j} \in \mathcal{C}$ , respectively.

Carry out the details of the proof of the following theorem (Theorem 4). If $\mathcal{M}_{j}, j \in I$ , are monotone classes, so is $\cap_{j \in I} \mathcal{M}_{j} = \left\{ A \subseteq \Omega ; A \in \mathcal{M}_{j}, j \in I\right\}$ .

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Problem 1.12

By means of an example, show that a monotone class need not be a field.

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Problem 1.13

Recall the following lemma (Lemma 2). Consider the rectangle $E = A \times B$ . Then, with `` $+$ '' denoting union of disjoint events,

$E^{c} = \left(A \times B^{c}\right) + \left(A^{c} \times \Omega_{2}\right) = \left(A^{c} \times B\right) + \left(\Omega_{1} \times B^{c}\right)$ . Consider the rectangles $E_{1} = A_{1} \times B_{1}$ , $E_{2} = A_{2} \times B_{2}$ . Then
$E_{1} \cap E_{2} = \left(A_{1} \cap A_{2}\right) \times \left(B_{1} \cap B_{2}\right)$ . Hence $E_{1} \cap E_{2} = \oslash$ if and only if at least one of the sets $A_{1} \cap A_{2}, B_{1} \cap B_{2}$ is $\oslash$ . Consider the rectangles $E_{1}, E_{2}$ as above, and the rectangles $F_{1} = A_{1}^{\prime } \times B_{1}^{\prime }, F_{2} = A_{2}^{\prime } \times B_{2}^{\prime }$ . Then
$\left(E_{1} \cap F_{1}\right) \cap \left(E_{2} \cap F_{2}\right) = \left[\left(A_{1} \cap A_{1}^{\prime }\right) \times \left(B_{1} \cap B_{1}^{\prime }\right)\right] \cap \left[\left(A_{2} \cap A_{2}^{\prime }\right) \times \left(B_{2} \cap B_{2}^{\prime }\right)\right]$ (by (ii)) $= \left[\left(A_{1} \cap A_{1}^{\prime }\right) \cap \left(A_{2} \cap A_{2}^{\prime }\right)\right] \times \left[\left(B_{1} \cap B_{1}^{\prime }\right) \cap \left(B_{2} \cap B_{2}^{\prime }\right)\right]$ (by (ii)) $= \left[\left(A_{1} \cap A_{2}\right) \cap \left(A_{1}^{\prime } \cap A_{2}^{\prime }\right)\right] \times \left[\left(B_{1} \cap B_{2}\right) \cap \left(B_{1}^{\prime } \cap B_{2}^{\prime }\right)\right]$ . Hence, the left-hand side is $\oslash$ if and only if at least one of $\left(A_{1} \cap A_{2}\right) \cap \left(A_{1}^{\prime } \cap A_{2}^{\prime }\right)$ , $\left(B_{1} \cap B_{2}\right) \cap \left(B_{1}^{\prime } \cap B_{2}^{\prime }\right)$ is $\oslash$ .

Carry out the details of the proof of Lemma 2.

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Problem 1.14

Let $\Omega_{1}, \Omega_{2}$ be two spaces and let $A, A_{i} \subseteq \Omega_{1}, B, B_{i} \subseteq \Omega_{2}, i = 1,2$ . Then show that

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(i)

$\left(A_{1} \times B_{1}\right)-\left(A_{2} \times B_{2}\right) = \left[\left(A_{1} \cap A_{2}\right) \times \left(B_{1}-B_{2}\right)\right]+\left[\left(A_{1}-A_{2}\right) \times B_{1}\right]$ .

(ii)

$A \times B = \oslash$ , if and only if at least one of $A, B$ is $\oslash$ .

(iii)

If $A_{i} \times B_{i}, i = 1,2$ are $\neq \varnothing$ , then $A_{1} \times B_{1} \subseteq A_{2} \times B_{2}$ , if and only if $A_{1} \subseteq A_{2}, B_{1} \subseteq B_{2}$ .

(iv)

If $A_{1} \times B_{1} = A_{2} \times B_{2} \neq \varnothing$ , then $A_{1} = A_{2}$ and $B_{1} = B_{2}$ .

(v)

Let $A \times B, A_{i} \times B_{i}, i = 1$ , 2 be $\neq \varnothing$ . Then $A \times B = \left(A_{1} \times B_{1}\right)+\left(A_{2} \times B_{2}\right)$ , if and only if $A = A_{1}+A_{2}$ and $B = B_{1} = B_{2}$ , or $A = A_{1} = A_{2}$ and $B = B_{1}+B_{2}$ .

Problem 1.15

(i)(ii)

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(i)

With $A \subseteq \Omega_{1}$ , and $B \subseteq \Omega_{2}$ , show that $A \times B = \oslash$ if and only if at least one of $A$ or $B$ is equal to $\oslash$ .

(ii)

With $A_{1}, A_{2} \subseteq \Omega_{1}$ and $B_{1}, B_{2} \subseteq \Omega_{2}$ , set $E_{1} = A_{1} \times B_{1}$ and $E_{2} = A_{2} \times B_{2}$ and assume that $E_{1}$ and $E_{2}$ are $\neq \varnothing$ . Then $E_{1} \subseteq E_{2}$ if and only if $A_{1} \subseteq A_{2}$ and $B_{1} \subseteq B_{2}$ . Explain why the assumption that $E_{1}$ and $E_{2}$ are $\neq \varnothing$ is essential.

Problem 1.16

(i)(ii)

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(i)

Let $A_{i} \subseteq \Omega_{i}, i = 1,2, \ldots , n$ , and set $E = A_{1} \times \cdots \times A_{n}$ . Then $E = \oslash$ if and only if at least one of $A_{i}, i = 1,2, \ldots , n$ , is $= \varnothing$ .

(ii)

If also $B_{i} \subseteq \Omega_{i}, i = 1,2, \ldots , n$ , and $F = B_{1} \times \cdots \times B_{n}$ , then show that $E \cap F = \left(A_{1} \times \cdots \times A_{n}\right) \cap \left(B_{1} \times \cdots \times B_{n}\right) = \left(A_{1} \cap B_{1}\right) \times \cdots \times \left(A_{n} \cap B_{n}\right)$ .

Problem 1.17

For $i = 1,2, \ldots , n$ , let $A_{i}, B_{i}, C_{i} \subseteq \Omega_{i}$ and set $E = A_{1} \times \cdots , \times A_{n}, F = B_{1} \times \cdots \times B_{n}, G = C_{1} \times \cdots \times C_{n}$ . Suppose that $E, F$ , and $G$ are all $\neq \varnothing$ and that $E = F+G$ . Then show that there exists a $j$ with $1 \leq j \leq n$ such that $A_{j} = B_{j}+C_{j}$ while $A_{i} = B_{i} = C_{i}$ for all $i \neq j$ .

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Problem 1.18

In reference to Theorem 7 , show that $\mathcal{C}$ is still a field, if $\mathcal{A}_{i}$ is replaced by a field $\mathcal{F}_{i}, i = 1,2$ .

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Problem 1.19

Consider the measurable spaces $\left(\Omega_{i}, \mathcal{A}_{i}\right), i = 1,2$ , and let $\mathcal{C}$ be the class of all countable sums of rectangles (unions of pairwise disjoint rectangles) in the product space $\Omega_{1} \times \Omega_{2}$ . Then by an example, show that $\mathcal{C}$ need not be a $\sigma$ -field.

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Remark.

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Compare it to Theorem 7 in this chapter.

Problem 1.20

Recall the following theorem (Theorem 10). Let $X$ be a mapping on $\Omega$ into $\Omega^{\prime }$ , where $\Omega^{\prime }$ is supplied with a $\sigma$ -field $\mathcal{A}^{\prime }$ , and define the class $\mathcal{C}$ of subsets of $\Omega$ as follows: $\mathcal{C} = X^{-1}\left(\mathcal{A}^{\prime }\right)$ . Then $\mathcal{C}$ is a $\sigma$ -field (i.e., the inverse image of a $\sigma$ -field is a $\sigma$ -field).

Carry out the details of the proof of Theorem 10.

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Problem 1.21

Recall the following theorem (Theorem 11). Let $X$ be a mapping on $\Omega$ into $\Omega^{\prime }$ , where $\Omega$ is supplied with a $\sigma$ -field $\mathcal{A}$ , and define the class $\mathcal{C}^{\prime }$ of subsets of $\Omega^{\prime }$ as follows: $\mathcal{C}^{\prime } = \left\{ A^{\prime } \subseteq \Omega^{\prime } ; X^{-1}\left(A^{\prime }\right) \in \mathcal{A}\right\}$ . Then $\mathcal{C}^{\prime }$ is a $\sigma$ -field.

Carry out the details of the proof of Theorem 11.

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Problem 1.22

Consider the mapping $X$ defined on $(\Omega , \mathcal{A})$ onto $\Omega^{\prime } = X(\Omega )$ , the image of $\Omega$ under $X$ , and let $\mathcal{C}^{\prime } \subseteq \mathcal{P}\left(\Omega^{\prime }\right)$ be defined as follows:

\mathcal{C}^{\prime } = \left\{ B \subseteq \Omega ^{\prime } ; \quad B = X(A), \quad A \in \mathcal{A}\right\}

Then, by means of an example, show that $\mathcal{C}^{\prime }$ need not be a $\sigma$ -field.

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Remark.

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Compare this result with Theorem 11 in this chapter.

Problem 1.23

Consider the measurable space $(\Omega , \mathcal{A})$ and let $X$ be defined by $\sum_{i = 1}^{n} \alpha_{i} I_{A_{i}}$ or $X = \sum_{i = 1}^{\infty } \alpha_{i} I_{A_{i}}$ , where $\alpha_{i} \in \mathbb {R}$ are distinct for all $i$ and $\left\{ A_{1}, \ldots , A_{n}\right\}$ or $\left\{ A_{i}, i \geq 1\right\}$ are partitions of $\Omega$ . Then show that $X$ is a r.v. (a simple r.v. and an elementary r.v., respectively) if and only if the partitions are measurable (i.e., $A_{i} \in \mathcal{A}$ for all $i$ ).

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Problem 1.24

If $X$ and $Y$ are mappings on $\Omega$ into $\mathbb {R}$ , show that

\begin{aligned} & \left\{ \omega \in \Omega ; \quad X(\omega )+Y(\omega ) < x\right\} = \bigcup _{r \in Q}[\left\{ \omega \in \Omega ; X(\omega ) < r\right\} \\ & \cap \left\{ \omega \in \Omega ; \quad Y(\omega ) < x-r\right\} ] \end{aligned}

where $Q$ is the set of rationals in $\mathbb {R}$ .

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Problem 1.25

If $X$ is a r.v. defined on the measurable space $(\Omega , \mathcal{A})$ , then $\left|X\right|$ is also a r.v. By an example, show that the converse need not be true.

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Problem 1.26

By a direct argument (that is, by using the definition of measurability), show that, if $X$ and $Y$ are r.v.s, then so are the mappings $X \pm Y, X Y$ , and $X / Y(Y \neq 0$ a.s.; i.e., $Y \neq 0$ with probability 1 , or $Y \neq 0$ almost surely).

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Problem 1.27

Recall the following theorem (Theorem 14). Let $f : \Omega \rightarrow \Omega^{\prime }$ be continuous, where $\Omega$ and $\Omega^{\prime }$ are supplied with topologies $\mathcal{T}$ and $\mathcal{T}^{\prime }$ , respectively, and let $\mathcal{T} \subseteq \mathcal{A}$ , $\mathcal{A}^{\prime } = \sigma \left(\mathcal{T}^{\prime }\right)$ . Then $f$ is measurable.

Carry out the details of the proof of the following corollary (the Corollary to Theorem 14). Let $X : (\Omega , \mathcal{A}) \rightarrow \left(\mathbb {R}^{n}, \mathcal{B}^{n}\right)$ be measurable and let $f : \left(\mathbb {R}^{n}, \mathcal{B}^{n}\right) \rightarrow \left(\mathbb {R}^{m}, \mathcal{B}^{m}\right)$ be continuous. Then $f(X) : \Omega \rightarrow \mathbb {R}^{m}$ is measurable (i.e., continuous functions of a random vector are random vectors), where $\mathcal{T}_{n}, \mathcal{T}_{m}$ denote the usual topologies on $\mathbb {R}^{n}, \mathbb {R}^{m}$ generating $\mathcal{B}^{n}, \mathcal{B}^{m}$ , respectively.

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Problem 1.28

If $X$ and $Y$ are r.v.s defined on $(\Omega , \mathcal{A})$ , show that $(X+Y)^{+} \leq X^{+}+Y^{+}$ and $(X+Y)^{-} \leq X^{-}+Y^{-}$ .

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Problem 1.29

(i)(ii)(iii)

Let $A_{1}, A_{2}, \ldots$ be arbitrary events in $(\Omega , \mathcal{A})$ , and define $B_{m}$ by: $B_{m} =$ " $A_{m}$ is the first event which occurs among the events $A_{1}, A_{2}, \ldots , " m \geq 1$ . Then

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(i)

Express $B_{m}$ in terms of $A_{n} \mathrm{~ s}, m \geq 1$ .

(ii)

Show that $B_{1}, B_{2}, \ldots$ are pairwise disjoint.

(iii)

Show that $\sum_{m = 1}^{\infty } B_{m} = \cup_{n = 1}^{\infty } A_{n}$ .

Problem 1.30

(i)(ii)

For a sequence of events $\left\{ A_{n}\right\} , n \geq 1$ , show that

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(i)

$\underline{\lim }_{n \rightarrow \infty } A_{n} = \left\{ \omega \in \Omega ; \quad \omega \in A_{n}\right.$ for all but finitely many $\left.n \mathrm{~ s}\right\}$ ,

(ii)

$\varlimsup_{n \rightarrow \infty } A_{n} = \left\{ \omega \in \Omega ; \omega \in A_{n}\right.$ for infinitely many $\left.n \mathrm{~ s}\right\}$ (to be denoted by ( $A_{n}$ i.o.) and read ( $A_{n}$ s occur infinitely often)).

Problem 1.31

If $A_{n}$ and $B_{n}$ are events such that $A_{n} \subseteq B_{n}, n \geq 1$ , then show that ( $A_{n}$ i.o.) $\subseteq$ ( $B_{n}$ i.o.).

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Problem 1.32

In $\mathbb {R}$ , let $Q$ be the set of rational numbers, and for $n = 1,2, \ldots$ , let $A_{n}$ be defined by

A_{n} = \left\{ r \in \left(1-\frac{1}{n+1}, 1+\frac{1}{n}\right) ; \quad r \in Q\right\} .

Examine whether or not the $\lim_{n \rightarrow \infty } A_{n}$ exists.

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Problem 1.33

In $\mathbb {R}$ , define the sets $A_{n}, n = 1,2, \ldots$ as follows:

A_{2 n-1} = \left[-1, \frac{1}{2 n-1}\right], \quad A_{2 n} = \left[0, \frac{1}{2 n}\right)

Examine whether or not the $\lim_{n \rightarrow \infty } A_{n}$ exists.

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Problem 1.34

(i)(ii)(iii)

Take $\Omega = \mathbb {R}$ , and let $\mathcal{A}_{n}$ be the $\sigma$ -field generated by the class $\{ [0,1),[1,2), \ldots$ , $[n-1, n)\} , n \geq 1$ . Then show that

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(i)

$\mathcal{A}_{n} \subseteq \mathcal{A}_{n+1}, n \geq 1$ , and indeed $\mathcal{A}_{n} \subset \mathcal{A}_{n+1}, n \geq 1$ .

(ii)

The class $\cup_{n = 1}^{\infty } \mathcal{A}_{n}$ is not a $\sigma$ -field.

(iii)

Describe explicitly $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ .

Problem 1.35

(i)(ii)

Let $A_{1}, \ldots , A_{n}$ be arbitrary subsets of an abstract set $\Omega$ , and let $A_{i}^{\prime }$ be either $A_{i}$ or $A_{i}^{c}, i = 1, \ldots , n$ . Define the class $\mathcal{C}$ of subsets of $\Omega$ as follows:

\mathcal{C} = \left\{ \text{ all unions of the intersections } A_{1}^{\prime } \cap \cdots \cap A_{n}^{\prime }\right\}

Then show that

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(i)

The class $\mathcal{C}$ is a field (generated by the sets $A_{1}, \ldots , A_{n}$ ).

(ii)

Compute the number of elements of $\mathcal{C}$ .

Problem 1.36

(i)(ii)(iii)

If $f: \Omega \rightarrow \Omega^{\prime }$ , then show that

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(i)

$f^{-1}[f(A)] \supseteq A, A \subseteq \Omega$ .

(ii)

$f\left[f^{-1}(B)\right] \subseteq B, B \subseteq \Omega^{\prime }$ .

(iii)

By concrete examples, show that the relations in (i) and (ii) may be strict.

Problem 1.37

(i)(ii)(iii)

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(i)

On the measurable space $(\Omega , \mathcal{A})$ , define the function $X$ as follows:

X(\omega ) = \left\{ \begin{aligned} -1 & \text{ on } A_{1} \\ 1 & \text{ on } A_{1}^{c} \cap A_{2} \\ 0 & \text{ on } A_{1}^{c} \cap A_{2}^{c} \end{aligned}\right.

where $A_{1}, A_{2} \in \mathcal{A}$ . Examine whether or not $X$ is a r.v.

(ii)

On the measurable space $(\Omega , \mathcal{A})$ with $\Omega = \left\{ a, b, c, d\right\}$ and $\mathcal{A} = \left\{ \varnothing ,\{ a, b$ ,c, d, right$$$$, define the function $X$ as follows: $X(a) = X(b) = -1, X(c) = 1, X(d) = 2$ . Examine whether or not $X$ is a r.v.

(iii)

If $\Omega = \left\{ -2,-1,0,1,2\right\}$ and $X$ is defined on $\Omega$ by $X(\omega ) = \omega$ , determine the field induced by $X$ and that induced by $X^{2}$ . Verify that the latter is contained in the former.

Problem 1.38

For a sequence of r.v.s $\left\{ X_{n}\right\} , n \geq 1$ , set $\mathcal{B}_{k} = \sigma \left(X_{k}, X_{k+1}, \ldots \right), k \geq 1$ . Then show that for every $k$ and $l$ with $k < l$ , it holds that $\mathcal{B}_{k} \supseteq \mathcal{B}_{l}$ .

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Problem 1.39

For the r.v.s $X_{1}, X_{2}, \ldots , X_{n}$ , set $S_{k} = \sum_{j = 1}^{k} X_{j}, k = 1, \ldots , n$ , and show that $\sigma \left(X_{1}, X_{2}, \ldots , X_{n}\right) = \sigma \left(S_{1}, S_{2}, \ldots , S_{n}\right)$ .

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Problem 1.40

For any set $B \subseteq \mathbb {R}$ , the set $B+c \stackrel{\text{ def }}{ = } B_{c}$ is defined by: $B_{c} = \{ y \in \mathbb {R}$ ; $y = x+c, x \in B\}$ . Then show that if $B$ is measurable, so is $B_{c}$ .

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Problem 1.41

(i)(ii)

Let $\Omega$ be an abstract set, and let $\mathcal{C}$ be an arbitrary nonempty class of subsets of $\Omega$ . Define the class $\mathcal{F}_{1}$ to consists of all members of $\mathcal{C}$ as well as all of their complements; i.e.,

\begin{aligned} \mathcal{F}_{1} & = \left\{ A \subseteq \Omega ; \quad A \in \mathcal{C} \text{ or } A = C^{c} \text{ with } C \in \mathcal{C}\right\} \\ & = \left\{ A \subseteq \Omega ; \quad A \in \mathcal{C} \text{ or } A^{c} \in \mathcal{C}\right\} = \mathcal{C} \cup \left\{ C^{c} ; \quad C \in \mathcal{C}\right\} , \end{aligned}

so that $\mathcal{F}_{1}$ is closed under complementation. Next, define the class $\mathcal{F}_{2}$ as follows:

\begin{align} \mathcal{F}_{2} = & \left\{ \text{ all finite intersections of members of } \mathcal{F}_{1}\right\} \\ = & \left\{ A \subseteq \Omega ; \quad A = A_{1} \cap \cdots \cap A_{m}, A_{i} \in \mathcal{F}_{1}, i = 1, \ldots , m, m \geq 1\right\} \end{align}

Also, define the class $\mathcal{F}_{3}$ by

\begin{align} \mathcal{F}_{3} = & \left\{ \text{ all finite unions of members of } \mathcal{F}_{2}\right\} \\ = & \left\{ A \subseteq \Omega ; \quad A = \bigcup _{i = 1}^{n} A_{i} \text{ with } A_{i} \in \mathcal{F}_{2}, i = 1, \ldots , n, n \geq 1\right\} \\ = & \left\{ A \subseteq \Omega ; \quad A = \bigcup _{i = 1}^{n} A_{i} \text{ with } A_{i} = A_{i}^{1} \cap \cdots \cap A_{i}^{m_{i}} A_{i}^{1}, \ldots , A_{i}^{m_{i}} \in \mathcal{F}_{1},, m_{i} \geq 1 \text{ integers } i = 1, \ldots , n, n \geq 1\right\} \end{align}

Set $\mathcal{F}_{3} = \mathcal{F}$ and show that

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(i)

$\mathcal{F}$ is a field.

(ii)

$\mathcal{F}$ is the field generated by $\mathcal{C}$ ; i.e., $\mathcal{F} = \mathcal{F}(\mathcal{C})$ .

Problem 1.42

Refer to Exercise 41 , and set $\mathcal{A}_{1} = \mathcal{F}_{1}$ . Then define the classes $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$ instead of $\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ , respectively, by replacing finite intersections and finite unions by countable intersections and countable unions, respectively. Set $\mathcal{A}_{3} = \mathcal{A}$ and examine whether or not $\mathcal{A}$ is a $\sigma$ -field.

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