# $\pi$-system, $\sigma$-field and Monotone class

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?

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 $$\{2\}=\Omega \setminus \{1\} \in \sigma (\mathcal P)$$, $$\{2\} \notin \mathcal M$$.