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

  1. $\mathcal{B_1} = \{k\mathbb{N} : k \in \mathbb{N} \}$
  2. $\mathcal{B_2} = \{k\mathbb{N} : k \ \ is \ a \ prime \}$

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.

  • $\begingroup$ For question 1, this is useful. $\endgroup$ May 2 at 9:32
  • $\begingroup$ I have no clue why you decided to vandalise your post, but please don't do it in the future. It doesn't work. $\endgroup$ May 5 at 13:26

Question 1

For $k \in \mathbb N$, you have

$$\{k + 1, k+2, \dots\} = \bigcup_{i \gt k} i \mathbb N$$ hence

$$\{k\} = k \mathbb N \setminus \bigcup_{i \gt k} i \mathbb N.$$ As each singleton belongs to $\sigma(\mathcal B_1)$, we get $\sigma(\mathcal B_1) = \mathcal P(\mathbb N)$ where $\mathcal P(\mathbb N)$ stands for the power set of $\mathbb N$.

Question 2 (to be completed)

Denote $\mathbb P$ the (countable) set of prime numbers and $\mathbb N_2 = \{2, 3,4,5, \dots \}$. We have

$$\mathbb N_2 = \bigcup_{p \in \mathbb P} p \mathbb N\in \sigma(\mathcal B_2)$$ and therefore $\mathbb N \in \sigma(\mathcal B_2)$.


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