Math Problems Vault

Problem 11.1.1

(a)(b)

Let $\Omega_{1}$ denote a nonempty set, and let $\mathcal{A}$ denote a nonempty collection of subsets of $\Omega_{1}$ . Let $\Omega_{2}$ denote a nonempty set, and let $\mathfrak {B}$ denote a nonempty collection of subsets of $\Omega_{2}$ .

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(a)

Let $B \subseteq \Omega_{2}$ be nonempty. Then $\sigma_{\Omega_{1} \times B}(\mathscr {A} \times \left\{ B\right\} )=\sigma (\mathscr {A}) \times \left\{ B\right\}$ .

(b)

$\sigma (\mathscr {A} \times \mathscr {B})=\sigma (\sigma (\mathscr {A}) \times \sigma (\mathscr {B}))$ .

Problem 11.1.2

Prove the claim in the Identification Lemma for the case where $P$ is a two-element set, which case is really the only one that we use.

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Remark.

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Identification Lemma: Let $\left\{ D_{p}\right\}_{p \in P}$ denote a disjoint family of nonempty subsets of $I$ with $\bigcup_{p \in P} D_{p}=I$ . Define $\Phi : \Omega \rightarrow \underset {p \in P}{\times } \Omega_{D_{p}}$ for every $\omega =(\omega_{i})_{i \in I} \in \Omega$ by the rule $\Phi (\omega )=\left((\omega_{i})_{i \in D_{p}}\right)_{p \in P}$ . Then: (a) $\Phi$ is one-to-one and onto $\underset {p \in P}{\times } \Omega_{D_{p}}$ . Let $\mathcal{F}^{*}$ denote the $\sigma$ -field of subsets of $\underset {p \in P}{\times } \Omega_{D_{p}}$ generated by sets of the form $\underset {p \in P}{\times } A_{D_{p}}$ , where $A_{D_{p}} \in \mathcal{F}_{D_{p}}$ for each $p \in P$ and $A_{D_{p}} \neq \Omega_{D_{p}}$ for at most finitely many $p \in P$ . (b) For every set in $\mathcal{F}$ there is a unique corresponding set in $\mathcal{F}^{*}$ , and vice versa; i.e., $\Phi$ establishes a one-to-one correspondence between $\mathcal{F}$ and $\mathcal{F}^{*}$ .

Problem 11.1.3

Prove the Identification Lemma in full generality for the case where $P$ is an arbitrary set.

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Remark.

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Identification Lemma: Let $\left\{ D_{p}\right\}_{p \in P}$ denote a disjoint family of nonempty subsets of $I$ with $\bigcup_{p \in P} D_{p}=I$ . Define $\Phi : \Omega \rightarrow \underset {p \in P}{\times } \Omega_{D_{p}}$ for every $\omega =(\omega_{i})_{i \in I} \in \Omega$ by the rule $\Phi (\omega )=\left((\omega_{i})_{i \in D_{p}}\right)_{p \in P}$ . Then: (a) $\Phi$ is one-to-one and onto $\underset {p \in P}{\times } \Omega_{D_{p}}$ . Let $\mathcal{F}^{*}$ denote the $\sigma$ -field of subsets of $\underset {p \in P}{\times } \Omega_{D_{p}}$ generated by sets of the form $\underset {p \in P}{\times } A_{D_{p}}$ , where $A_{D_{p}} \in \mathcal{F}_{D_{p}}$ for each $p \in P$ and $A_{D_{p}} \neq \Omega_{D_{p}}$ for at most finitely many $p \in P$ . (b) For every set in $\mathcal{F}$ there is a unique corresponding set in $\mathcal{F}^{*}$ , and vice versa; i.e., $\Phi$ establishes a one-to-one correspondence between $\mathcal{F}$ and $\mathcal{F}^{*}$ .

Problem 11.1.4

Show that $\mathscr {E}_{F}$ is a semiring on $\Omega_{F}$ .

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Problem 11.1.5

Let $\mathcal{A}$ denote a semiring on $\Omega_{1}$ , and let $\mathfrak {B}$ denote a semiring on $\Omega_{2}$ . Then $\mathcal{A} \times \mathcal{B}$ is a semiring on $\Omega_{1} \times \Omega_{2}$ .

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