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I'm trying to solve exercise 1.9.42 in the Resnick book. This exercise involves the concept of monotonous class, $\pi$-system and $\sigma$-algebra.

Assume $\mathcal{P}$ is a $\pi-system$ and $\mathcal{M}$ is a monotone class. Show $\mathcal{P}\subset\mathcal{M}$ does not imply $\sigma(\mathcal{P})\subset\mathcal{M}$.

I started looking for examples in which I have a set that is a $\pi-system$ and a monotonous class, but that the $\sigma(\mathcal{P})$ is not a monotone class. But I doubt if this is really what the exercise calls for, as it would make no sense to find a $\sigma$-algebra that does not belong to the monotone class, since every $\sigma$-algebra is a monotone class. Anyway I thought of the following example:

$\Omega$={1,2,3,4}

P={{1,2,3,4},{2,3,4},{3,4},{4}}

It's a $\pi$ system?

{1,2,3,4}$\bigcap${2,3,4}={2,3,4} ok

{2,3,4}$\bigcap${3,4} ok

{2,3,4}$\bigcap${4} ok

{3,4}$\bigcap{4}$ ok

It's a monotone classe?

{1,2,3,4}$\supset${2,3,4}$\supset${3,4}$\supset${4}

Now I would need to check the sigma algebra ...

That makes sense?

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I think you msiunderstood the question. You have to come up with two families $\mathcal P$ and $\mathcal M$.

Let $\Omega=\{1,2\}, \mathcal P=\{\{1\}\}$ and $\mathcal M=\{\{1\}, \{1,2\}\}$. Then $\mathcal P$ is a $\pi-$ system and $\mathcal M$ is a monotone class. But $\{2\}=\Omega \setminus \{1\} \in \sigma (\mathcal P)$, $\{2\} \notin \mathcal M$.

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  • $\begingroup$ Very thanks for your reply. $\endgroup$ Commented Jan 9, 2021 at 13:21

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