Math Problems Vault

Problem 1.4.1

Show directly that $\sigma \left(\mathscr {A}_{3}\right)=\sigma \left(\mathscr {A}_{3}^{*}\right), \sigma \left(\mathscr {A}_{4}\right)=\sigma \left(\mathscr {A}_{7}\right)$ , and $\sigma \left(\mathcal{A}_{4}^{*}\right)=\sigma \left(\mathcal{A}_{10}\right)$ .

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Remark.

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Notation: $\mathscr {A}_{3}=$ intervals of the form $[x, \infty )$ , $\mathscr {A}_{4}=$ intervals of the form $(x, \infty )$ , $\mathscr {A}_{7}=$ intervals of the form $[a, b)$ , and $\mathscr {A}_{10}=$ closed subsets of $\mathbb {R}$ . An asterisk on a collection (e.g. $\mathscr {A}_{3}^{*}$ , $\mathcal{A}_{4}^{*}$ ) denotes the same form of interval restricted to rational endpoints.

Problem 1.4.2

All amc subsets of $\mathbb {R}$ are Borel sets. All subsets of $\mathbb {R}$ that differ from a Borel set by at most countably many points are Borel sets. That is, if the symmetric difference $C \Delta B$ is amc and $B \in \mathfrak {B}$ , then $C \in \mathfrak {B}$ .

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Problem 1.4.3

The Borel $\sigma$ -field on $(0,1]$ is denoted by $\mathscr {B}_{(0,1]}$ and is defined as the $\sigma$ -field on $(0,1]$ generated by the rsc subintervals of $(0,1] . \mathscr {B}_{(0,1]}$ may be equivalently defined by $\left\{ B \cap (0,1] \mid B \in \mathscr {B}\right\}$ .

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Problem 1.4.4

$\mathscr {B}$ is generated by the compact subsets of $\mathbb {R}$ .

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Problem 1.4.5

(a)(b)(c)(d)

$\mathscr {B}$ is not generated by the following:

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(a)

Any finite collection of subsets of $\mathbb {R}$ .

(b)

The collection of real singletons.

(c)

The collection of all finite subsets of $\mathbb {R}$ .

(d)

The collection of all amc subsets of $\mathbb {R}$ .

Problem 1.4.6

The representations (1)-(5) give equivalent formulations of $\bar{\mathscr {B}}$ .

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Remark.

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Definition: the extended Borel $\sigma$ -field on $\overline{\mathbb {R}}$ is denoted by $\bar{\mathscr {B}}$ , and is defined by $\bar{\mathscr {B}} \equiv \sigma \left(\left\{ (a, b]: a, b \in \overline{\mathbb {R}}, a<b\right\} \right)$ . Sets in $\bar{\mathscr {B}}$ are called extended Borel sets.

Just as different authors use different definitions for $\mathscr {B}$ , so too $\bar{\mathscr {B}}$ has various definitions. Some other definitions are (1) the class of sets that are $\mathscr {B}$ -sets or are $\mathscr {B}$ -sets enlarged by one or both of $\pm \infty$ , (2) the $\sigma$ -field on $\overline{\mathbb {R}}$ generated by the class of intervals of the form $(a, b)$ , $[-\infty , a)$ , and $(b,+\infty ]$ , where $a, b \in \mathbb {R}$ , (3) the $\sigma$ -field on $\overline{\mathbb {R}}$ generated by intervals of the form $[-\infty , x)$ , $x \in \mathbb {R}$ , (4) $\sigma \left(\mathscr {B} \cup \left\{ \left\{ -\infty \right\} ,\left\{ +\infty \right\} \right\} \right)$ , and (5) the $\sigma$ -field on $\overline{\mathbb {R}}$ generated by the open subsets of $\overline{\mathbb {R}}$ , sets of the form $A$ , $A \cup (x, \infty ]$ , $A \cup [-\infty , x)$ , and $A \cup [-\infty , x) \cup (y,+\infty ]$ , where $A$ is an open subset of $\mathbb {R}$ and $x, y \in \mathbb {R}$ .

Claim 2 (Section 1.4): $\mathscr {B}=\sigma \left(\mathcal{A}_{1}\right)=\cdots =\sigma \left(\mathcal{A}_{10}\right)=\sigma \left(\mathcal{A}_{1}^{*}\right)=\cdots =\sigma \left(\mathcal{A}_{8}^{*}\right)$ , where $\mathcal{A}_{1}, \ldots , \mathcal{A}_{10}$ and $\mathcal{A}_{1}^{*}, \ldots , \mathcal{A}_{8}^{*}$ are the classes of intervals defined in the remark to problem 1.4.1 (with $\mathcal{A}_{n}^{*}$ the same form as $\mathcal{A}_{n}$ but with rational endpoints, for $n=1, \ldots , 8$ ).

The natural extension of Claim 2 to $\overline{\mathbb {R}}$ holds in the extended Borel setting: $\bar{\mathscr {B}}$ is generated by any of the analogously-extended versions of $\mathcal{A}_{1}, \ldots , \mathcal{A}_{10}, \mathcal{A}_{1}^{*}, \ldots , \mathcal{A}_{8}^{*}$ as well.