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Let $\mathbb{N} = \{1,2,3,4,\dots \}$ and define the sets $A_k \subset \mathbb{N}$ by $$ A_k = \{k,2k,3k,\dots \} $$ for $k = 1,2,\dots$. We denote by $\mathcal{H}$ the collection $\{A_1, A_2, \dots \} \cup \emptyset$.

Show that $\sigma(\mathcal{H})$ (the sigma-algebra generated by $\mathcal{H}$) is equal to $\mathcal{P}(\mathbb{N})$ (the power set of $\mathbb{N}$).

We know that $A_i^c \in \sigma(\mathcal{P})$ and finite intersections are again in the sigma-algebra. I tried to construct singletons out of these operations because if they are in $\sigma(\mathcal{H})$ then could construct the powerset by taking unions (I think) but I couldn't do it. I also found out that $A_i \cap A_j = A_{lcm(i,j)} \in \mathcal{H}$. So that $\mathcal{H}$ is stable under finite intersections. But now I'm stuck.

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  • $\begingroup$ Hint: Can you show the singleton $\{47\}$ is in $\sigma(\mathcal H)$? $\endgroup$
    – GEdgar
    Commented Feb 7, 2016 at 14:05
  • $\begingroup$ Is it true because $47$ is a prime number the set $\{47 \}$ is only in $A_1$ and $A_{47}$. So that for example with $2$, because it is prime as well, we can construct $\{2 \} = A_2 \cap (\cup_{3}^{\infty} A_i)$? $\endgroup$
    – Nescrio
    Commented Feb 7, 2016 at 14:17
  • $\begingroup$ Next, how about $\{k\}$ for composite $k$? $\endgroup$
    – GEdgar
    Commented Feb 7, 2016 at 14:20
  • $\begingroup$ I think we can extend this technique to composite numbers as well. $\{6\}\in A_6 = A_2 \cap A_3$. But isn't it true that $\{6\} = A_6 \cap (\cup_{i=7}^{\infty} A_i)$. If that's true we constructed the singletons and we're done! $\endgroup$
    – Nescrio
    Commented Feb 7, 2016 at 14:37
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    $\begingroup$ Perhaps you want $\{6\} = A_6 \setminus (\cup_{i=7}^{\infty} A_i)$. $\endgroup$
    – GEdgar
    Commented Feb 7, 2016 at 14:41

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To answer my own question:

We can construct the singletons as follows: $$ \{i\} = A_i \backslash \cup_{t = i+1}^{\infty}A_t, \quad \forall i \in \mathbb{N}. $$ Now that we know $\sigma(\mathcal{H})$ contains all the singletons and therefore we know that this sigma-algebra is equal to the powerset of $\mathbb{N}$, because we can construct it by taking unions of singletons.

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