Let $X$ be a set, and let $\mathcal{F} \subseteq \mathcal{P}(X)$ be an infinite family of subset of $X$ of cardinality $\kappa$. Show that $\langle \mathcal{F} \rangle$ has cardinality at most $\kappa^{\aleph_0}$. In particular, Borel $\sigma$-algebra has cardinality at most $2^{\aleph_0}$.

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An Introduction to Measure Theory

Exercise 1.4.16 (Cardinality of a $\sigma$-algebra)

Let $X$ be a set, and let $ℱ \subseteq 𝒫 (X)$ be an infinite family of subset of $X$ of cardinality $κ$ . Show that $⟨ ℱ ⟩$ has cardinality at most $κ^{ℵ_{0}}$ . In particular, Borel $σ$ -algebra has cardinality at most $2^{ℵ_{0}}$ .

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eluthingol

2020-07-02 00:00

Exercise 1.4.16 (Cardinality of a $\sigma$-algebra)

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