Let $\mathbb{N} = \{1,2,3,\dots\}$ and $k\mathbb{N} =\{k,2k,3k,\dots\}$.
What is the $\sigma$-field generated by the following collections
- $\mathcal{B_1} = \{k\mathbb{N} : k \in \mathbb{N} \}$
- $\mathcal{B_2} = \{k\mathbb{N} : k \ \ is \ a \ prime \}$
I can prove if the description of $\sigma$-field is given that it'll be generated by some given collection but I can't guess the other way. Any hints is appreciated.