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I would like to define a $\mathbb{N}$-valued random variable to define the $\sigma$-algebra generated by this random variable. In my course, we've define only $\sigma$-algebras on borelians of $\mathbb{R}$ ( by the set $\{ X^{-1}(A), A \in B(\mathbb{R}) \}$ for the real case). So what would be borelian of $\mathbb{N}$? Is that just disjoint elements of $\mathbb{N}$ ? Thanks

William Elliot
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wainwain
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    All $1$-element subsets of $\mathbb N$ are Borel sets, but so are all the (at most) countable unions of those, implying that every subset of $\mathbb N$ is a Borel set. –  Nov 24 '20 at 22:25
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    @StinkingBishop Heck, every subset of $\mathbb{N}$ is open, at least with the usual topology. – Noah Schweber Nov 25 '20 at 00:02

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