Transfinite construction of a generated sigma-ring

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Problem 1.3.7 from Vestrup E.M. "The Theory of Measures and Integration" [Ves03]More problems in Section 1.3 "$\sigma$-Fields"

(a)(b)(c)

This problem continues problem 1.3.6. Let $\mathcal{A}$ denote a collection of subsets of some set $\Omega$ , and let $S(\mathcal{A})$ denote the minimal $\sigma$ -ring that contains $\mathcal{A}$ .

(a)

For any collection $\mathcal{C}$ of sets, let $\mathcal{C}^{*}$ denote the collection of all countable unions of differences of $\mathcal{C}$ -sets. Define $\mathcal{A}_{0}=\mathcal{A}$ , and for any ordinal $\alpha \succ 0$ , define $\mathcal{A}_{\alpha }=\left(\bigcup_{0 \preceq \beta \prec \alpha } \mathcal{A}_{\beta }\right)^{*}$ . Then $0 \preceq \alpha \prec \beta$ implies $\mathcal{A} \subseteq \mathcal{A}_{\alpha } \subseteq \mathcal{A}_{\beta } \subseteq S(\mathcal{A})$ .

(b)

Letting $\omega$ denote the first uncountable ordinal, show that $S(\mathcal{A})=\bigcup_{0 \preceq \alpha \prec \omega } \mathcal{A}_{\alpha }$ .

(c)

If $\operatorname {card}(\mathcal{A}) \leq \mathfrak {c}$ , then $\operatorname {card}(S(\mathcal{A})) \leq \mathfrak {c}$ .

References

[Ves03] Vestrup E.M. The Theory of Measures and Integration. Wiley-Interscience, 2003.