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Let $\Omega=\mathbb{R} $ and $\mathcal{S}=\{\{x\}:x \in \Omega \}$

a) $\mathbb{N} \in \sigma(\mathcal{S})$?

b)Prove that $]0,1[ \not \in \sigma(\mathcal{S})$)

With the question a) I haven´t problem being that $\sigma(\mathcal{S})$ is a $\sigma$-algebra then Union of natural numbers $\in$ $\sigma(\mathcal{S})$ then $\mathbb{N} \in \sigma(\mathcal{S})$ ,

but b) I think it makes accounting sets but I do not know how propose

logan_diaz
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1 Answers1

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Show that $\mathcal A = \{ X \subseteq \mathbb R \mid X \text{ is countable or } \mathbb R \setminus X \text{ is countable} \}$ is a $\sigma$-algebra. As $\mathcal S \subseteq \mathcal A$, the claim follows.

Stefan Mesken
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