# $\sigma$-fields generated by multiples of natural numbers

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.

• For question 1, this is useful. May 2 at 9:32
• I have no clue why you decided to vandalise your post, but please don't do it in the future. It doesn't work. 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 $$\mathbb N \in \sigma(\mathcal B_2)$$.